cotx 1=cos2x/1+tanx+sin^2x 1/2sin2x

Prove that:
$\frac{1-\cos 2x+\sin 2x}{1+\cos 2x+\sin 2\mathrm{x}}=\tan x$

Solve the following equations:
(i) $\cos x+\cos 2x+\cos 3x=0$
(ii) $\cos x+\cos 3x-\cos 2x=0$
(iii) $\sin x+\sin 5x=\sin 3x$
(iv) $\cos x \cos 2x \cos 3x=\frac{1}{4}$
(v) $\cos x+\sin x=\cos 2x+\sin 2x$
(vi) $\sin x+\sin 2x+\sin 3=0$
(vii) $\sin x+\sin 2x+\sin 3x+\sin 4x=0$
(viii) $\sin 3x-\sin x=4 {\cos }^2 x-2$
(ix) $\sin 2x-\sin 4x+\sin 6x=0$

Prove that
cos4x+cos3x+cos2xsin4x+sin3x+sin2x equals

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Find the common roots of the equations
cos2x+sin2x=cotx
and 2cos2x+cos22x=1