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NCERT Exercise 3.3 Question 6 \cos \left ( \frac{\pi}{4} – x \right )\cos \left ( \frac{\pi}{4} – y \right ) – \sin \left ( \frac{\pi}{4} – x \right )\sin \left ( \frac{\pi}{4} – y \right ) = \sin (x+y)

Question 7 \frac{\tan \left ( \frac{\pi}{4} + x \right )}{\tan \left ( \frac{\pi}{4} – x \right )} = \left ( \frac{1 + \tan x}{1 – \tan x} \right ) ^2

Question 8 \frac{cos(\pi+x) \cos(-x)}{\sin (\pi – x) \cos \left ( \frac{\pi}{2} + x \right )} = \cot^2 x

Question 9 \cos \left ( \frac{3\pi}{2} + x \right ) cos (2 \pi + x) \left [ \cot \left ( \frac{3\pi}{2} – x \right ) + \cot (2 \pi + x) \right ] = 1

Question 10 \sin (n + 1)x \sin (n + 2)x + \cos (n + 1)x \cos (n + 2)x = \cos x

Derivation for
\sin (A+B)+\sin (A-B) = 2 \sin A \cos B
\sin (A+B)-\sin (A-B) = 2 \cos A \sin B

\cos (A+B)+\cos (A-B) = 2 \cos A \cos B
\cos (A+B)-\cos (A-B) = – 2 \sin A \sin B

Question 11 \cos \left ( \frac{3\pi}{4} + x \right ) – \cos \left ( \frac{3\pi}{4} – x \right ) = – \sqrt{2} \sin x

Derivation for twice angles
\sin 2x = 2 \sin x \cos x = \frac{2 \tan x}{1 + \tan^2 x}
\cos 2x = \cos^2 x – \sin^2 x = 1 – 2 \sin^2 x = 2 \cos^2 x – 1 = \frac{1-\tan^2 x}{1 + \tan^2 x}

Question 24 \cos 4x = 1 – 8 \sin^2 x \cos^2 x

Question 23 \tan 4x = \frac{4 \tan x (1 – \tan^2 x)}{1-6\tan^2 x + \tan^4 x}