$1. \sin(x+\frac{\pi}{4} )^{4}=\frac{1}{4}\cos x^{2}-\cos x^{4}$
$2. 2\sin x+2\sin2x=\cot x+1$
$3. \tan x^{2}-\frac{\tan x}{\cot3x}=2$
$4. 2\cos x^{2}+2\sqrt{3}\sin x\cos x+1=3sinx+3\sqrt{3}\cos x$
$5. \frac{1}{\tan x+\cot2x}=\frac{\sqrt{2}(\cos x-\sin x)}{\cot x-1}$
$6. \sin x^{6}+\cos x^{6}=\sin x^{4}+\cos x^{4}+1+\cos2x$
$7. (\tan x-\tan2x)(\tan x-\sin2x)=3$