$a)\,2\tan x+\cot x=2\sin2x+\dfrac{1}{\sin2x}\\ b)\,\sqrt{3}\sin\left(3x-\dfrac{\pi}{5}\right)+2\sin\left(8x-\dfrac{\pi}{3}\right)=2\sin\left(2x+\dfrac{11\pi}{15}\right)+3\cos\left(3x-\dfrac{\pi}{5}\right)\\c)\,\sin^2x+\sin x\cos3x+\cos^23x=\dfrac{3}{4}\\d)\,\left(\tan x\cot2x-1\right)\sin\left(4x+\dfrac{\pi}{2}\right)=-\dfrac{1}{2}\left(\sin^4x+\cos^4x\right)$...