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

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

\(\frac{1}{\sqrt{2}}\cot x+\frac{\sin2x}{\sin x+\cos x}=2\sin\left(x+\frac{\pi}{2}\right)\)

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

3sinx+2cosx=3sin2x

\(2\sin\left(2x-\frac{\pi}{6}\right)+4sinx=0\)

\(2\left(\sin^4x+cos^4x\right)=2sin2x-1\)

\(2cosxcos2x=1+cos2x+cos3x\)

\(\tan x-3\cot x=4\left(\sin x+\sqrt{3}\cos x\right)\)

\(\sin^4x+\cos^4\left(x+\frac{\pi}{4}\right)=\frac{1}{4}\)