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tan 10° · tan 80° = ?
tan 10° · tan 80° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Complementary Property Shortcut Method): Step 1: Direct rule: Agar $A + B = 90^\circ$ ho, toh $\tan A \cdot \tan B = 1$ hota hai. Step 2: Yahan angles check karein: $$10^\circ + 80^\circ = 90^\circ$$ Step 3: Isliye product direct $1$ hoga: $$\tan 10^\circ \cdot \tan 80^\circ = 1$$ Correct option (a) 1 hai. Method 2 (Step-by-Step Conversion Method): $$\tan 80^\circ = \tan(90^\circ - 10^\circ) = \cot 10^\circ = \frac{1}{\tan 10^\circ}$$ $$\tan 10^\circ \times \frac{1}{\tan 10^\circ} = 1$$ Sahi uttar (a) hai.
tan 39° · tan 43° · tan 47° · tan 51° = ?
tan 39° · tan 43° · tan 47° · tan 51° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Complementary Pairs Shortcut Method): Step 1: Un angles ke pairs banayein jinka sum $90^\circ$ banta hai: - Pair 1: $(39^\circ + 51^\circ = 90^\circ) \implies \tan 39^\circ \cdot \tan 51^\circ = 1$ - Pair 2: $(43^\circ + 47^\circ = 90^\circ) \implies \tan 43^\circ \cdot \tan 47^\circ = 1$ Step 2: Multiply karein: $$\text{Total} = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Conversion Method): $$\tan 39^\circ \cdot \tan 43^\circ \cdot \cot 43^\circ \cdot \cot 39^\circ = (\tan 39^\circ\cot 39^\circ) \cdot (\tan 43^\circ\cot 43^\circ) = 1 \times 1 = 1$$ Sahi uttar (a) hai.
tan 10° · tan 20° · tan 30° · tan 40° · tan 50° · tan 70° · tan 80° = ?
tan 10° · tan 20° · tan 30° · tan 40° · tan 50° · tan 70° · tan 80° का मान क्या होगा?
(A) (a) √3
(B) (b) 1/√3
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: B) (b) 1/√3Explanation: Method 1 (Pairing & Standard Value Method): Step 1: $90^\circ$ sum wale pairs alag karein: - $\tan 10^\circ \cdot \tan 80^\circ = 1$ - $\tan 20^\circ \cdot \tan 70^\circ = 1$ - $\tan 40^\circ \cdot \tan 50^\circ = 1$ Step 2: Beech me keval standard angle $\tan 30^\circ$ bachta hai: $$\text{Product} = 1 \times 1 \times 1 \times \tan 30^\circ = \tan 30^\circ$$ Step 3: $\tan 30^\circ$ ki value $\frac{1}{\sqrt{3}}$ hoti hai: $$\text{Value} = \frac{1}{\sqrt{3}}$$ Correct option (b) 1/√3 hai. Method 2 (Direct Inspection Method): Teen pairs cancel hokar 1 banenge, single bacha term $\tan 30^\circ = \frac{1}{\sqrt{3}}$ answer hoga. Sahi uttar (b) hai.
tan 1° · tan 2° · tan 3° ......... tan 88° · tan 89° = ?
tan 1° · tan 2° · tan 3° ......... tan 88° · tan 89° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Series Pairing Method): Step 1: Total terms 1 se 89 tak $= 89$ terms hain. Step 2: Shuru aur aakhiri se complementary pairs banate hain: - $(\tan 1^\circ \cdot \tan 89^\circ) = 1$ - $(\tan 2^\circ \cdot \tan 88^\circ) = 1$ - $(\tan 44^\circ \cdot \tan 46^\circ) = 1$ Kul 44 pairs banenge jinki value $1 \times 1 \times \dots = 1$ hogi. Step 3: Exactly middle term $\tan 45^\circ$ bachega, jiska man $1$ hota hai: $$\text{Total} = (1)^{44} \times \tan 45^\circ = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Symmetry Rule): Pure product me saare terms ek dusre ke reciprocal bankar cancel ho jate hain aur center term 1 rehta hai, so overall product = 1. Sahi uttar (a) hai.
tan 1° · tan 3° · tan 5° ......... tan 87° · tan 89° = ?
tan 1° · tan 3° · tan 5° ......... tan 87° · tan 89° का मान क्या होगा?
(A) (a) √3
(B) (b) 1/√3
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: C) (c) 1Explanation: Method 1 (Odd Angles Series Pairing Method): Step 1: Series me keval odd angles hain (1, 3, 5, ..., 89). Total terms $= \frac{89 - 1}{2} + 1 = 45$ terms. Step 2: $90^\circ$ sum wale pairs banate hain: - $\tan 1^\circ \cdot \tan 89^\circ = 1$ - $\tan 3^\circ \cdot \tan 87^\circ = 1$ - $\tan 43^\circ \cdot \tan 47^\circ = 1$ Kul 22 pairs banenge jo sabhi 1 ho jayenge. Step 3: Beech ka term $\tan 45^\circ$ bachega: $$\text{Product} = (1)^{22} \times \tan 45^\circ = 1 \times 1 = 1$$ Correct option (c) 1 hai. Method 2 (Direct Inspection): Har angle ka complementary partner mojood hai, aur center term $\tan 45^\circ = 1$ hai, isliye answer 1 aayega. Sahi uttar (c) hai.
tan 5° · tan 10° · tan 15° ......... tan 80° · tan 85° = ?
tan 5° · tan 10° · tan 15° ......... tan 80° · tan 85° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Multiples of 5 Pairing Method): Step 1: Angles 5 ke multiples hain 5 se 85 tak. Total terms $= \frac{85}{5} = 17$ terms. Step 2: $90^\circ$ sum wale pairs banayein: - $\tan 5^\circ \cdot \tan 85^\circ = 1$ - $\tan 10^\circ \cdot \tan 80^\circ = 1$ - $\tan 15^\circ \cdot \tan 75^\circ = 1$ - $\tan 40^\circ \cdot \tan 50^\circ = 1$ Aise 8 pairs banenge. Step 3: Central term $\tan 45^\circ$ single rahega: $$\text{Product} = (1)^8 \times \tan 45^\circ = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Reciprocal Multiplication Method): $$\prod_{k=1}^8 (\tan(5k)^\circ \cdot \cot(5k)^\circ) \times \tan 45^\circ = 1^8 \times 1 = 1$$ Sahi uttar (a) hai.
If tan A · tan B = 1, then A + B = ?
यदि tan A · tan B = 1 है, तो A + B का मान क्या होगा?
(A) (a) 60°
(B) (b) 90°
(C) (c) 30°
(D) (d) 120°
✅ Answer & Explanation
Sahi jawab: B) (b) 90°Explanation: Method 1 (Direct Trigonometric Identity Method): Step 1: Given equation: $\tan A \cdot \tan B = 1$. $$\tan A = \frac{1}{\tan B} = \cot B$$ Step 2: $\cot B$ ko tangent me convert karein: $$\tan A = \tan(90^\circ - B)$$ Step 3: Angles equate karein: $$A = 90^\circ - B \implies A + B = 90^\circ$$ Correct option (b) 90° hai. Method 2 (Value Testing Method): $A = 45^\circ, B = 45^\circ$ assume karein: $\tan 45^\circ \cdot \tan 45^\circ = 1 \times 1 = 1$. $$A + B = 45^\circ + 45^\circ = 90^\circ$$ Sahi uttar (b) hai.
If tan θ · tan 2θ = 1, then sin 3θ = ?
यदि tan θ · tan 2θ = 1 है, तो sin 3θ का मान क्या होगा?
(A) (a) 1/√2
(B) (b) 1/2
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: C) (c) 1Explanation: Method 1 (Angle Sum Property Method): Step 1: Rule: Jab $\tan A \cdot \tan B = 1$ ho, tab $A + B = 90^\circ$ hota hai. Step 2: Angles add karein: $$\theta + 2\theta = 90^\circ \implies 3\theta = 90^\circ$$ Step 3: $\sin 3\theta$ evaluate karein: $$\sin 3\theta = \sin 90^\circ = 1$$ Correct option (c) 1 hai. Method 2 (Value-Putting Method): $3\theta = 90^\circ \implies \theta = 30^\circ$. $$\tan 30^\circ \cdot \tan 60^\circ = \frac{1}{\sqrt{3}} \times \sqrt{3} = 1 \quad (\text{Satisfied})$$ $$\sin(3 \times 30^\circ) = \sin 90^\circ = 1$$ Sahi uttar (c) hai.
If tan 3θ · tan 6θ = 1, then sin 3θ + cos 3θ = ?
यदि tan 3θ · tan 6θ = 1 है, तो sin 3θ + cos 3θ का मान क्या होगा?
(A) (a) (1 + √3)/2
(B) (b) (1 - √3)/2
(C) (c) √3
(D) (d) 1/√3
✅ Answer & Explanation
Sahi jawab: A) (a) (1 + √3)/2Explanation: Method 1 (Angle Extraction & Evaluation Method): Step 1: Property: $\tan A \cdot \tan B = 1 \implies A + B = 90^\circ$. $$3\theta + 6\theta = 90^\circ \implies 9\theta = 90^\circ \implies \theta = 10^\circ$$ Step 2: $3\theta$ ki value nikalte hain: $$3\theta = 3 \times 10^\circ = 30^\circ$$ Step 3: Target expression $\sin 3\theta + \cos 3\theta$ me angle rakhein: $$\sin 30^\circ + \cos 30^\circ = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}$$ Correct option (a) (1 + √3)/2 hai. Method 2 (Direct Step Calculation): $$3\theta = 30^\circ \implies \sin 30^\circ + \cos 30^\circ = 0.5 + 0.866 = \frac{1 + \sqrt{3}}{2}$$ Sahi uttar (a) hai.
If tan(x + y) · tan(x - y) = 1, then sin x + cos x = ?
यदि tan(x + y) · tan(x - y) = 1 है, तो sin x + cos x का मान क्या होगा?
(A) (a) 1
(B) (b) √2
(C) (c) 0
(D) (d) - 1
✅ Answer & Explanation
Sahi jawab: B) (b) √2Explanation: Method 1 (Angle Sum Property Method): Step 1: Property: $\tan A \cdot \tan B = 1 \implies A + B = 90^\circ$. $$(x + y) + (x - y) = 90^\circ$$ $$2x = 90^\circ \implies x = 45^\circ$$ Step 2: Target expression me $x = 45^\circ$ substitute karein: $$\sin 45^\circ + \cos 45^\circ = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$ Correct option (b) √2 hai. Method 2 (Value Substitution Method): $x = 45^\circ, y = 15^\circ$ consider karein: $\tan 60^\circ \cdot \tan 30^\circ = \sqrt{3} \times \frac{1}{\sqrt{3}} = 1$. $$\sin 45^\circ + \cos 45^\circ = \frac{2}{\sqrt{2}} = \sqrt{2}$$ Sahi uttar (b) hai.
If tan(α + 2β) · tan(α - 2β) = 1, then tan α + cot α = ?
यदि tan(α + 2β) · tan(α - 2β) = 1 है, तो tan α + cot α का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: D) (d) 2Explanation: Method 1 (Angle Addition Shortcut Method): Step 1: $\tan A \cdot \tan B = 1 \implies A + B = 90^\circ$. $$(\alpha + 2\beta) + (\alpha - 2\beta) = 90^\circ$$ $$2\alpha = 90^\circ \implies \alpha = 45^\circ$$ Step 2: $\tan\alpha + \cot\alpha$ evaluate karein: $$\tan 45^\circ + \cot 45^\circ = 1 + 1 = 2$$ Correct option (d) 2 hai. Method 2 (Direct Identity Method): $$\tan\alpha + \cot\alpha = \frac{2}{\sin 2\alpha} = \frac{2}{\sin 90^\circ} = \frac{2}{1} = 2$$ Sahi uttar (d) hai.
If tan(x + 3y) · tan(x - 3y) = 1, then cos x = ?
यदि tan(x + 3y) · tan(x - 3y) = 1 है, तो cos x का मान क्या होगा?
(A) (a) 1/√2
(B) (b) 1/2
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: A) (a) 1/√2Explanation: Method 1 (Angle Sum Property Method): Step 1: Dono angles ka sum $90^\circ$ hona chahiye: $$(x + 3y) + (x - 3y) = 90^\circ$$ $$2x = 90^\circ \implies x = 45^\circ$$ Step 2: $\cos x$ ki value evaluate karein: $$\cos 45^\circ = \frac{1}{\sqrt{2}}$$ Correct option (a) 1/√2 hai. Method 2 (Testing with y = 0): $y = 0 \implies \tan x \cdot \tan x = 1 \implies \tan^2 x = 1 \implies x = 45^\circ$. $$\cos 45^\circ = \frac{1}{\sqrt{2}}$$ Sahi uttar (a) hai.
If tan 37° · tan θ = 1, then θ = ?
यदि tan 37° · tan θ = 1 है, तो θ का मान क्या होगा?
(A) (a) 53°
(B) (b) 43°
(C) (c) 63°
(D) (d) 90°
✅ Answer & Explanation
Sahi jawab: A) (a) 53°Explanation: Method 1 (Complementary Angle Formula Method): Step 1: $\tan A \cdot \tan B = 1 \implies A + B = 90^\circ$. Step 2: Value substitute karein: $$37^\circ + \theta = 90^\circ$$ $$\theta = 90^\circ - 37^\circ = 53^\circ$$ Correct option (a) 53° hai. Method 2 (Cotangent Method): $$\tan\theta = \frac{1}{\tan 37^\circ} = \cot 37^\circ = \tan(90^\circ - 37^\circ) = \tan 53^\circ \implies \theta = 53^\circ$$ Sahi uttar (a) hai.
If tan(π/12) · tan θ = 1, then θ = ?
यदि tan(π/12) · tan θ = 1 है, तो θ का मान क्या होगा?
(A) (a) 3π/10
(B) (b) 5π/12
(C) (c) 7π/12
(D) (d) π/12
✅ Answer & Explanation
Sahi jawab: B) (b) 5π/12Explanation: Method 1 (Radian Complementary Angle Method): Step 1: Radians me complementary sum $\frac{\pi}{2}$ hota hai: $$\frac{\pi}{12} + \theta = \frac{\pi}{2}$$ Step 2: $\theta$ solve karein: $$\theta = \frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi - \pi}{12} = \frac{5\pi}{12}$$ Correct option (b) 5π/12 hai. Method 2 (Degree Conversion Method): $$\frac{\pi}{12} = \frac{180^\circ}{12} = 15^\circ$$ $$\theta = 90^\circ - 15^\circ = 75^\circ$$ $$75^\circ = 75 \times \frac{\pi}{180} = \frac{5\pi}{12}$$ Sahi uttar (b) hai.
tan(π/12) · tan(5π/12) = ?
tan(π/12) · tan(5π/12) का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Sum of Angles Method): Step 1: Dono angles ko add karke dekhein: $$\frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} = 90^\circ$$ Step 2: Chuki angles complementary hain, isliye product 1 hoga: $$\tan\left(\frac{\pi}{12}\right) \cdot \tan\left(\frac{5\pi}{12}\right) = 1$$ Correct option (a) 1 hai. Method 2 (Standard Numerical Values Method): $$\frac{\pi}{12} = 15^\circ, \quad \frac{5\pi}{12} = 75^\circ$$ $$\tan 15^\circ = 2 - \sqrt{3}, \quad \tan 75^\circ = 2 + \sqrt{3}$$ $$\tan 15^\circ \cdot \tan 75^\circ = (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1$$ Sahi uttar (a) hai.
tan(π/10) · tan(2π/10) · tan(3π/10) · tan(4π/10) = ?
tan(π/10) · tan(2π/10) · tan(3π/10) · tan(4π/10) का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Radian Pair Grouping Method): Step 1: Complementary pairs (sum = $\frac{5\pi}{10} = \frac{\pi}{2}$) banate hain: - Pair 1: $\tan(\pi/10) \cdot \tan(4\pi/10) = 1$ (chuki $\frac{\pi}{10} + \frac{4\pi}{10} = \frac{\pi}{2}$) - Pair 2: $\tan(2\pi/10) \cdot \tan(3\pi/10) = 1$ (chuki $\frac{2\pi}{10} + \frac{3\pi}{10} = \frac{\pi}{2}$) Step 2: Multiply karein: $$\text{Product} = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Degree Conversion Method): $$\frac{\pi}{10} = 18^\circ \implies \tan 18^\circ \cdot \tan 36^\circ \cdot \tan 54^\circ \cdot \tan 72^\circ$$ $$= (\tan 18^\circ \cdot \tan 72^\circ) \times (\tan 36^\circ \cdot \tan 54^\circ) = 1 \times 1 = 1$$ Sahi uttar (a) hai.
tan(π/20) · tan(3π/20) · tan(7π/20) · tan(9π/20) = ?
tan(π/20) · tan(3π/20) · tan(7π/20) · tan(9π/20) का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Pairing Method for π/2 Sum): Step 1: Pairs banayein jinka sum $\frac{10\pi}{20} = \frac{\pi}{2}$ ($90^\circ$) ho: - $\frac{\pi}{20} + \frac{9\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2} \implies \tan(\pi/20) \cdot \tan(9\pi/20) = 1$ - $\frac{3\pi}{20} + \frac{7\pi}{20} = \frac{10\pi}{20} = \frac{\pi}{2} \implies \tan(3\pi/20) \cdot \tan(7\pi/20) = 1$ Step 2: Product evaluate karein: $$\text{Total} = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Degree Equivalent Method): $$\frac{\pi}{20} = 9^\circ \implies \tan 9^\circ \cdot \tan 27^\circ \cdot \tan 63^\circ \cdot \tan 81^\circ$$ $$= (\tan 9^\circ \cdot \tan 81^\circ) \times (\tan 27^\circ \cdot \tan 63^\circ) = 1 \times 1 = 1$$ Sahi uttar (a) hai.
If tanA · tanB = 1, then sin²A + sin²B = ?
यदि tanA · tanB = 1 है, तो sin²A + sin²B का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Complementary Angle Shortcut Method): Step 1: Formula: Agar $\tan A \cdot \tan B = 1$ ho, toh $A + B = 90^\circ$ hota hai. Step 2: $B = 90^\circ - A$ substitute karein: $$\sin B = \sin(90^\circ - A) = \cos A$$ $$\sin^2 B = \cos^2 A$$ Step 3: Target expression me rakhein: $$\sin^2 A + \sin^2 B = \sin^2 A + \cos^2 A = 1$$ Correct option (a) 1 hai. Method 2 (Value-Putting Method): $A = 45^\circ, B = 45^\circ$ assume karein (chuki $\tan 45^\circ \cdot \tan 45^\circ = 1$): $$\sin^2 45^\circ + \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1$$ Sahi uttar (a) hai.
If tanα · tanβ = 1, then (sinα / cosβ) = ?
यदि tanα · tanβ = 1 है, तो (sinα / cosβ) का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Complementary Angle Ratio Method): Step 1: $\tan\alpha \cdot \tan\beta = 1 \implies \alpha + \beta = 90^\circ \implies \beta = 90^\circ - \alpha$. Step 2: Denominator ko convert karein: $$\cos\beta = \cos(90^\circ - \alpha) = \sin\alpha$$ Step 3: Fraction evaluate karein: $$\frac{\sin\alpha}{\cos\beta} = \frac{\sin\alpha}{\sin\alpha} = 1$$ Correct option (a) 1 hai. Method 2 (Value-Putting Method): $\alpha = 30^\circ, \beta = 60^\circ$ lein (chuki $\tan 30^\circ \cdot \tan 60^\circ = 1$): $$\frac{\sin 30^\circ}{\cos 60^\circ} = \frac{1/2}{1/2} = 1$$ Sahi uttar (a) hai.
If tanx · tany = 1, then (secx / cosecy) = ?
यदि tanx · tany = 1 है, तो (secx / cosecy) का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Complementary Equivalence Method): Step 1: $\tan x \cdot \tan y = 1 \implies x + y = 90^\circ \implies y = 90^\circ - x$. Step 2: Denominator ko simplify karein: $$\csc y = \csc(90^\circ - x) = \sec x$$ Step 3: Ratio nikalte hain: $$\frac{\sec x}{\csc y} = \frac{\sec x}{\sec x} = 1$$ Correct option (a) 1 hai. Method 2 (Value-Putting Method): $x = 45^\circ, y = 45^\circ$ assume karein: $$\frac{\sec 45^\circ}{\csc 45^\circ} = \frac{\sqrt{2}}{\sqrt{2}} = 1$$ Sahi uttar (a) hai.
cot 10° · cot 80° = ?
cot 10° · cot 80° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Direct Complementary Property Method): Step 1: Rule: Agar $A + B = 90^\circ$ ho, toh $\cot A \cdot \cot B = 1$ hota hai. Step 2: Angles check karein: $$10^\circ + 80^\circ = 90^\circ$$ Step 3: Isliye product direct $1$ hoga: $$\cot 10^\circ \cdot \cot 80^\circ = 1$$ Correct option (a) 1 hai. Method 2 (Conversion Method): $$\cot 80^\circ = \cot(90^\circ - 10^\circ) = \tan 10^\circ$$ $$\cot 10^\circ \times \tan 10^\circ = 1$$ Sahi uttar (a) hai.
cot 39° · cot 43° · cot 47° · cot 51° = ?
cot 39° · cot 43° · cot 47° · cot 51° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Pairing Complementary Angles Method): Step 1: $90^\circ$ yog wale pairs banate hain: - Pair 1: $(39^\circ + 51^\circ = 90^\circ) \implies \cot 39^\circ \cdot \cot 51^\circ = 1$ - Pair 2: $(43^\circ + 47^\circ = 90^\circ) \implies \cot 43^\circ \cdot \cot 47^\circ = 1$ Step 2: Multiply karein: $$\text{Product} = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Step-by-Step Conversion Method): $$\cot 39^\circ \cdot \cot 43^\circ \cdot \tan 43^\circ \cdot \tan 39^\circ = (\cot 39^\circ\tan 39^\circ)(\cot 43^\circ\tan 43^\circ) = 1 \times 1 = 1$$ Sahi uttar (a) hai.
cot 10° · cot 20° · cot 30° · cot 40° · cot 50° · cot 70° · cot 80° = ?
cot 10° · cot 20° · cot 30° · cot 40° · cot 50° · cot 70° · cot 80° का मान क्या होगा?
(A) (a) √3
(B) (b) 1/√3
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: A) (a) √3Explanation: Method 1 (Pairing & Standard Value Method): Step 1: Complementary pairs alag karte hain: - $\cot 10^\circ \cdot \cot 80^\circ = 1$ - $\cot 20^\circ \cdot \cot 70^\circ = 1$ - $\cot 40^\circ \cdot \cot 50^\circ = 1$ Step 2: Beech me keval standard angle $\cot 30^\circ$ bachega: $$\text{Total} = 1 \times 1 \times 1 \times \cot 30^\circ = \cot 30^\circ$$ Step 3: $\cot 30^\circ = \sqrt{3}$ hota hai: $$\text{Value} = \sqrt{3}$$ Correct option (a) √3 hai. Method 2 (Direct Inspection): Teeno pairs cancel hokar 1 banenge, single bacha term $\cot 30^\circ = \sqrt{3}$ hi final answer hoga. Sahi uttar (a) hai.
cot 1° · cot 2° · cot 3° ......... cot 88° · cot 89° = ?
cot 1° · cot 2° · cot 3° ......... cot 88° · cot 89° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Symmetric Series Pairing Method): Step 1: Total terms 1 se 89 tak $= 89$ terms hain. Step 2: Complementary pairs banate hain: - $(\cot 1^\circ \cdot \cot 89^\circ) = 1$ - $(\cot 2^\circ \cdot \cot 88^\circ) = 1$ - $(\cot 44^\circ \cdot \cot 46^\circ) = 1$ Kul 44 pairs 1 ban jayenge. Step 3: Center term $\cot 45^\circ$ bachega, jiska man $1$ hota hai: $$\text{Total} = (1)^{44} \times \cot 45^\circ = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Symmetry Rule): Sabhi reciprocal terms cancel hokar $\cot 45^\circ = 1$ chhodte hain, so overall product = 1. Sahi uttar (a) hai.
cot 1° · cot 3° · cot 5° ......... cot 87° · cot 89° = ?
cot 1° · cot 3° · cot 5° ......... cot 87° · cot 89° का मान क्या होगा?
(A) (a) √3
(B) (b) 1/√3
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: C) (c) 1Explanation: Method 1 (Odd Angles Series Method): Step 1: Series me keval odd angles hain (total 45 terms). Step 2: $90^\circ$ sum wale 22 pairs banenge: - $(\cot 1^\circ \cdot \cot 89^\circ) = 1$ - $(\cot 3^\circ \cdot \cot 87^\circ) = 1$ Step 3: Exactly middle term $\cot 45^\circ$ bachega: $$\text{Product} = (1)^{22} \times \cot 45^\circ = 1 \times 1 = 1$$ Correct option (c) 1 hai. Method 2 (Direct Inspection): Har term ka complementary partner mojood hai aur $\cot 45^\circ = 1$ hai, isliye answer 1 hoga. Sahi uttar (c) hai.
cot 5° · cot 10° · cot 15° ...... cot 80° · cot 85° = ?
cot 5° · cot 10° · cot 15° ...... cot 80° · cot 85° का मान क्या होगा?
(A) (a) 1
(B) (b) - 1
(C) (c) 0
(D) (d) 2
✅ Answer & Explanation
Sahi jawab: A) (a) 1Explanation: Method 1 (Multiples of 5 Pairing Method): Step 1: 5 ke multiples wale kul 17 terms hain. Step 2: $90^\circ$ sum wale 8 pairs banenge: - $(\cot 5^\circ \cdot \cot 85^\circ) = 1$ - $(\cot 10^\circ \cdot \cot 80^\circ) = 1$ - $\dots$ Step 3: Beech ka akela term $\cot 45^\circ$ bachta hai: $$\text{Product} = 1^8 \times \cot 45^\circ = 1 \times 1 = 1$$ Correct option (a) 1 hai. Method 2 (Tangent Inversion Method): $$\prod_{k=1}^8 (\cot 5k^\circ \cdot \tan 5k^\circ) \times \cot 45^\circ = 1 \times 1 = 1$$ Sahi uttar (a) hai.
If cotA · cotB = 1, then A + B = ?
यदि cotA · cotB = 1 है, तो A + B का मान क्या होगा?
(A) (a) 60°
(B) (b) 90°
(C) (c) 30°
(D) (d) 120°
✅ Answer & Explanation
Sahi jawab: B) (b) 90°Explanation: Method 1 (Trigonometric Property Method): Step 1: Given: $\cot A \cdot \cot B = 1$. $$\cot A = \frac{1}{\cot B} = \tan B$$ Step 2: $\tan B$ ko cotangent me convert karein: $$\cot A = \cot(90^\circ - B)$$ Step 3: Angles equate karein: $$A = 90^\circ - B \implies A + B = 90^\circ$$ Correct option (b) 90° hai. Method 2 (Value Testing Method): $A = 45^\circ, B = 45^\circ \implies \cot 45^\circ \cdot \cot 45^\circ = 1 \times 1 = 1$. $$A + B = 45^\circ + 45^\circ = 90^\circ$$ Sahi uttar (b) hai.
If cotθ · cot 2θ = 1, then sin 3θ = ?
यदि cotθ · cot 2θ = 1 है, तो sin 3θ का मान क्या होगा?
(A) (a) 1/√2
(B) (b) 1/2
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: C) (c) 1Explanation: Method 1 (Angle Sum Property Method): Step 1: Jab $\cot A \cdot \cot B = 1$ ho, tab $A + B = 90^\circ$ hota hai. Step 2: Angles add karein: $$\theta + 2\theta = 90^\circ \implies 3\theta = 90^\circ$$ Step 3: Target value $\sin 3\theta$ nikalte hain: $$\sin 3\theta = \sin 90^\circ = 1$$ Correct option (c) 1 hai. Method 2 (Angle Determination Method): $$3\theta = 90^\circ \implies \theta = 30^\circ$$ $$\cot 30^\circ \cdot \cot 60^\circ = \sqrt{3} \times \frac{1}{\sqrt{3}} = 1 \quad (\text{Satisfied})$$ $$\sin(3 \times 30^\circ) = \sin 90^\circ = 1$$ Sahi uttar (c) hai.
If cot 3θ · cot 6θ = 1, then sin 3θ + cos 3θ = ?
यदि cot 3θ · cot 6θ = 1 है, तो sin 3θ + cos 3θ का मान क्या होगा?
(A) (a) (1 + √3)/2
(B) (b) (1 - √3)/2
(C) (c) √3
(D) (d) 1/√3
✅ Answer & Explanation
Sahi jawab: A) (a) (1 + √3)/2Explanation: Method 1 (Angle Extraction Method): Step 1: Property: $\cot A \cdot \cot B = 1 \implies A + B = 90^\circ$. $$3\theta + 6\theta = 90^\circ \implies 9\theta = 90^\circ \implies \theta = 10^\circ$$ Step 2: $3\theta = 3 \times 10^\circ = 30^\circ$. Step 3: Value substitute karein: $$\sin 3\theta + \cos 3\theta = \sin 30^\circ + \cos 30^\circ = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}$$ Correct option (a) (1 + √3)/2 hai. Method 2 (Direct Calculation): $$\frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}$$ Sahi uttar (a) hai.
If cot(x + y) · cot(x - y) = 1, then sin x + cos x = ?
यदि cot(x + y) · cot(x - y) = 1 है, तो sin x + cos x का मान क्या होगा?
(A) (a) 1
(B) (b) √2
(C) (c) 0
(D) (d) - 1
✅ Answer & Explanation
Sahi jawab: B) (b) √2Explanation: Method 1 (Complementary Sum Method): Step 1: Rule: $\cot A \cdot \cot B = 1 \implies A + B = 90^\circ$. $$(x + y) + (x - y) = 90^\circ$$ $$2x = 90^\circ \implies x = 45^\circ$$ Step 2: Value expression me substitute karein: $$\sin 45^\circ + \cos 45^\circ = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$ Correct option (b) √2 hai. Method 2 (Value Testing Method): $x = 45^\circ, y = 15^\circ$ assume karein: $\cot 60^\circ \cdot \cot 30^\circ = \frac{1}{\sqrt{3}} \times \sqrt{3} = 1$. $$\sin 45^\circ + \cos 45^\circ = \frac{2}{\sqrt{2}} = \sqrt{2}$$ Sahi uttar (b) hai.