NCERT Solutions
Class 11 - Mathematics - Chapter 3: TRIGONOMETRIC FUNCTIONS

EXERCISE 3.3

Prove that \( \sin^2\left(\dfrac{\pi}{6}\right) + \cos^2\left(\dfrac{\pi}{3}\right) - \tan^2\left(\dfrac{\pi}{4}\right) = -\tfrac{1}{2} \).

Prove that \( 2\sin^2\left(\dfrac{\pi}{6}\right) + \csc^2\left(\dfrac{7\pi}{6}\right) \cos^2\left(\dfrac{\pi}{3}\right) = \tfrac{3}{2} \).

Prove that \( \cot^2\left(\dfrac{\pi}{6}\right) + \csc\left(\dfrac{5\pi}{6}\right) + 3\tan^2\left(\dfrac{\pi}{6}\right) = 6 \).

Prove that \( 2\sin^2\left(\dfrac{3\pi}{4}\right) + 2\cos^2\left(\dfrac{\pi}{4}\right) + 2\sec^2\left(\dfrac{\pi}{3}\right) = 10 \).

Find the value of:

(i) \( \sin 75^{\circ} \)

(ii) \( \tan 15^{\circ} \)

Prove that \( \cos\left(\dfrac{\pi}{4} - x\right) \cos\left(\dfrac{\pi}{4} - y\right) - \sin\left(\dfrac{\pi}{4} - x\right) \sin\left(\dfrac{\pi}{4} - y\right) = \sin(x + y) \).

Prove that \( \dfrac{\tan\left(\dfrac{\pi}{4} + x\right)}{\tan\left(\dfrac{\pi}{4} - x\right)} = \left( \dfrac{1 + \tan x}{1 - \tan x} \right)^2 \).

Prove that \( \dfrac{\cos(\pi + x) \cos(-x)}{\sin(\pi - x) \cos\left(\dfrac{\pi}{2} + x\right)} = \cot^2 x \).

Prove that \( \cos\left(\dfrac{3\pi}{2} + x\right) \cos(2\pi + x) \left[ \cot\left(\dfrac{3\pi}{2} - x\right) + \cot(2\pi + x) \right] = 1 \).

Prove that \( \sin(n + 1)x \sin(n + 2)x + \cos(n + 1)x \cos(n + 2)x = \cos x \).

Prove that \( \cos\left(\dfrac{3\pi}{4} + x\right) - \cos\left(\dfrac{3\pi}{4} - x\right) = -\sqrt{2} \sin x \).

Prove that \( \sin^2 6x - \sin^2 4x = \sin 2x \sin 10x \).

Prove that \( \cos^2 2x - \cos^2 6x = \sin 4x \sin 8x \).

Prove that \( \sin 2x + 2 \sin 4x + \sin 6x = 4 \cos^2 x \sin 4x \).

Prove that \( \cot 4x \left( \sin 5x + \sin 3x \right) = \cot x \left( \sin 5x - \sin 3x \right) \).

Prove that \( \dfrac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\dfrac{\sin 2x}{\cos 10x} \).

Prove that \( \dfrac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x \).

Prove that \( \dfrac{\sin x - \sin y}{\cos x + \cos y} = \tan \dfrac{x - y}{2} \).

Prove that \( \dfrac{\sin x + \sin 3x}{\cos x + \cos 3x} = \tan 2x \).

Prove that \( \dfrac{\sin x - \sin 3x}{\sin^2 x - \cos^2 x} = 2 \sin x \).

Prove that \( \dfrac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x \).

Prove that \( \cot x \cot 2x - \cot 2x \cot 3x - \cot 3x \cot x = 1 \).

Prove that \( \tan 4x = \dfrac{4 \tan x \left(1 - \tan^2 x\right)}{1 - 6 \tan^2 x + \tan^4 x} \).

Prove that \( \cos 4x = 1 - 8 \sin^2 x \cos^2 x \).

Prove that \( \cos 6x = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1 \).