Question

Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3x cos x
(ii) 2 cos 3x sin 2x
(iii) 2 sin 4x sin 3x
(iv) 2 cos 7x cos 3x

Answer 1

(i)
$$\begin{gathered} 2\left(\sin 3x\right)\left(\cos x\right) \\ =\sin \left(3x+x\right)+\sin \left(3x-x\right)\left[\because 2\sin AcosB=\sin (A+B)+\sin (A-B)\right] \\ =\sin 4x+\sin 2x \end{gathered}$$

(ii)
$$\begin{gathered} 2\left(\cos 3x\right)\left(\sin 2x\right) \\ =\sin \left(3x+2x\right)-\sin \left(3x-2x\right)\left[\because 2\cos A\sin B=\sin (A+B)-\sin (A-B)\right] \\ =\sin 5x-\sin x \end{gathered}$$

(iii)
$$\begin{gathered} 2\left(\sin 4x\right)\left(\sin 3x\right) \\ =\cos \left(4x-3x\right)-\cos \left(4x+3x\right)\left[\because 2\sin A\sin B=\cos (A-B)-\cos (A+B)\right] \\ =\cos x-\cos 7x \end{gathered}$$

(iv)
$$\begin{gathered} 2\left(\cos 7x\right)\left(\cos 3x\right) \\ =\cos \left(7x+3x\right)+\cos \left(7x-3x\right)\left[\because 2\cos A\cos B=\cos (A+B)+\cos (A-B)\right] \\ =\cos 10x+\cos 4x \end{gathered}$$