Express Each as the Sum or Difference of Sines and Cosines
Using product-to-sum identities, express the following trigonometric products as sums or differences of sines and cosines.
Product-to-Sum Identities Used
\[
2\sin A \cos B = \sin(A+B) + \sin(A-B)
\]
\[
2\cos A \sin B = \sin(A+B) – \sin(A-B)
\]
\[
2\sin A \sin B = \cos(A-B) – \cos(A+B)
\]
\[
2\cos A \cos B = \cos(A+B) + \cos(A-B)
\]
(i) \(2\sin 3x \cos x\)
\[
2\sin 3x \cos x
= \sin(3x+x)+\sin(3x-x)
\]
\[
= \sin 4x+\sin 2x
\]
(ii) \(2\cos 3x \sin 2x\)
\[
2\cos 3x \sin 2x
= \sin(3x+2x)-\sin(3x-2x)
\]
\[
= \sin 5x-\sin x
\]
(iii) \(2\sin 4x \sin 3x\)
\[
2\sin 4x \sin 3x
= \cos(4x-3x)-\cos(4x+3x)
\]
\[
= \cos x-\cos 7x
\]
(iv) \(2\cos 7x \cos 3x\)
\[
2\cos 7x \cos 3x
= \cos(7x+3x)+\cos(7x-3x)
\]
\[
= \cos 10x+\cos 4x
\]
Final Answers
\[
\begin{aligned}
(i)\;& 2\sin 3x \cos x = \sin 4x+\sin 2x \\
(ii)\;& 2\cos 3x \sin 2x = \sin 5x-\sin x \\
(iii)\;& 2\sin 4x \sin 3x = \cos x-\cos 7x \\
(iv)\;& 2\cos 7x \cos 3x = \cos 10x+\cos 4x
\end{aligned}
\]