We must embrace pain and burn it as fuel for our journey.

Chapter 3 Miscellaneous Exercise Question 1 \( 2 \cos \frac{\pi}{13} \cos \frac{9\pi}{13} + \cos \frac{3\pi}{13}\cos \frac{5\pi}{13} = 0\)

Question 2 \( (\sin 3x + \sin x) \sin x + (\cos 3x – \cos x) \cos x = 0\)

Question 3 \( (\cos x + \cos y)^2 + (\sin x – \sin y)^2 = 4 \cos^2 \frac{x+y}{2}\)

Question 4 \( (\cos x – \cos y)^2 + (\sin x – \sin y)^2 = 4 \sin^2 \frac{x-y}{2}\)

Question 5 \( \sin x + \sin 3x + \sin 5x + \sin 7x = 4 \cos x \cos 2x \sin 4x\)

Question 6 \( \frac{(\sin 7x + \sin 5x) + (\sin 9x + \sin 3x)}{(\cos 7x + \cos 5x) + (\cos 9x + \cos 3x)} = \tan 6x\)

Question 7 \( \sin 3x + \sin 2x – \sin x = 4\sin x \cos \frac{x}{2} \cos \frac{3x}{2}\)

Example 25 If \( \sin x = \frac{3}{5},  \cos y = \frac{-12}{13}\), where x and y both lie in second quadrant, find the value of \( \sin (x+y)\)

Example 26 Prove that
\( \cos 2x \cos \frac{x}{2} – \cos 3x \cos \frac{9x}{2} = \sin 5x \sin \frac{5x}{2}\)