Question

The value of $\cos \left(15°\right)\cos \left(7\frac{1°}{2}\right)\sin \left(7\frac{1°}{2}\right)$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{8}$
• C. $\frac{1}{4}$
• D. $\frac{1}{16}$

Answer 1

The correct option is B

$\frac{1}{8}$

Explanation for correct option

Given trigonometric function is $\cos \left(15°\right)\cos \left(7\frac{1°}{2}\right)\sin \left(7\frac{1°}{2}\right)$

$$\begin{gathered} \therefore \cos \left(15°\right)\cos \left(7\frac{1°}{2}\right)\sin \left(7\frac{1°}{2}\right) \\ =\frac{1}{2}\left[\cos \left(15°\right)·2·\cos \left(\frac{15°}{2}\right)\sin \left(\frac{15°}{2}\right)\right] \\ =\frac{1}{2}\left[\cos \left(15°\right)·\sin \left(15°\right)\right]\left[\because 2\sin \left(A\right)\cos \left(A\right)=\sin \left(2A\right)\right] \\ =\frac{1}{4}\left[2\cos \left(15°\right)·\sin \left(15°\right)\right] \\ =\frac{1}{4}\left[\sin \left(30°\right)\right]\left[\because 2\sin \left(A\right)\cos \left(A\right)=\sin \left(2A\right)\right] \\ =\frac{1}{4}·\frac{1}{2}\left[\because \left[\sin \left(30°\right)\right]=\frac{1}{2}\right] \\ =\frac{1}{8} \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{B}\right)$