$a)\,2\left(\cos x+\sqrt{3}\sin x\right)\cos x=\cos x-\sqrt{3}\sin x+1\\b)\,\dfrac{\sin x\sin2x+2\sin x\cos^2x+\sin x\cos x}{\cos\left(x-\dfrac{\pi}{4}\right)}=\sqrt{6}\cos2x\\c)\,\sin\left(\dfrac{3\pi}{10}-\dfrac{x}{2}\right)=\dfrac{1}{2}\sin\left(\dfrac{\pi}{10}+\dfrac{3x}{2}\right)$...