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Q (8) Prove the following

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

 

Q (9) Prove the following

\small \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

Q (10) Prove the following

\small \sin (n+1)x\sin(n+2)x + \cos(n+1)x\cos(n+2)x =\cos x

Q (11) Prove the following

\small \cos \left ( \frac{3\pi }{4}+x \right ) - \cos\left ( \frac{3\pi }{4} -x\right ) = -\sqrt{2} \sin x
 

Q (12) Prove the following

\small \sin^{2}6x - \sin^{2}4x = \sin2x\sin10x

Q (13) Prove the following

\small \cos^{2}2x - \cos^{2}6x = \sin4x\sin8x

Q (14) Prove the following

\small \sin2x +2\sin4x + \sin6x = 4\cos^{2}x\sin4x

Q (15) Prove the following

\small \cot4x(\sin5x + \sin3x) = \cot x(\sin5x - \sin3x)

Q (16) Prove the following

\small \frac{\cos 9x - \cos 5x}{\sin17x - \sin3x} = -\frac{\sin2x}{\cos10x}

Q (17) Prove the following

\small \frac{\sin5x + \sin3x}{\cos5x + \cos3x} = \tan4x

Q (18) Prove the following

\small \frac{\sin x - \sin y}{\cos x+\cos y} = \tan \frac{(x-y)}{2}

Q (19)  Prove the following

\small \frac{\sin x + \sin 3x}{\cos x + \cos3x} = \tan2x

Q (20)  Prove the following

\small \frac{\sin x - \sin 3x}{\sin^{2}x-\cos^{2}x} = 2\sin x

Q (21)  Prove the following

\small \frac{\cos 4x + \cos 3x + \cos 2x}{\sin 4x + \sin 3x + \sin 2x} = \cot 3x

Find the principal and general solutions of the following equations:

Q (1)   \small \tan x = \sqrt{3}

Find the principal and general solutions of the following equations:
 
Q (2)  \small \sec x = 2

Find the principal and general solutions of the following equations:

Q (3)   \small \cot x = - \sqrt{3}

Find the principal and general solutions of the following equations:

Q (4)  \small cosec x = -2

Find the general solution for each of the following equation 

Q (5)  \small \cos 4x = \cos 2x

Find the general solution of the following equation 

Q (6)  \small \cos 3x + \cos x -\cos 2x = 0

Q (7) Find the general solution of the following equation 

\small \sin 2x + \cos x = 0

Q (8)  Find the general solution of the following equation 

   \small \sec^{2}2x = 1 - \tan2x

Q (9): Find the general solution of the following equation

   \small \sin x + \sin 3x + \sin 5x = 0

Q (1)  Prove that

\small 2\cos\frac{\pi }{13}\cos\frac{9\pi }{13}+\cos\frac{3\pi }{13}+\cos\frac{5\pi }{13}=0

Q (2)   Prove that 

\small (\sin 3x + \sin x)\sin x + (\cos 3x - \cos x )\cos x = 0

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