Express each of the following as the sum or difference of sines and cosines:

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

Solution:

(i) 2 sin 3x cos x

$=\sin (3 x+x)+\sin (3 x-x) \quad[\because 2 \sin A \cos B=\sin (A+B)+\sin (A-B)]$

$=\sin 4 x+\sin 2 x$

(ii) 2 cos 3x sin 2x

$=\sin (3 x+2 x)-\sin (3 x-2 x) \quad[\because 2 \cos A \sin B=\sin (A+B)-\sin (A-B)]$

$=\sin 5 x-\sin x$

(iii) 2 sin 4x sin 3x

$=\cos (4 x-3 x)-\cos (4 x+3 x) \quad[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)]$

$=\cos x-\cos 7 x$

(iv) 2 cos 7x cos 3x

$=\cos (7 x+3 x)+\cos (7 x-3 x) \quad[\because 2 \cos A \cos B=\cos (A+B)+\cos (A-B)]$

$=\cos 10 x+\cos 4 x$

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