Question

If $\cos A=\frac{3}{4}$ then $32\sin \frac{A}{2}\sin \frac{5A}{2}$

• A. $10$
• B. $-11$
• C. $11$
• D. $7$

Answer 1

The correct option is D $7$

$2\sin \frac{A}{2}\sin \frac{5A}{2}$

$=\cos (\frac{A}{2}-\frac{5A}{2})-\cos (\frac{A}{2}+\frac{5A}{2})$

$=\cos 2A-\cos 3A-----\left(1\right)$

Now, $\cos A=\frac{3}{4}$

$\therefore \cos 2A=2{\cos }^{2}A-1=2\frac{9}{16}-1=\frac{1}{8}$

$\cos 3A=4{\cos }^{3}A-3\cos A=4\times \frac{27}{64}-3\times \frac{3}{4}$

$=\frac{27}{16}-\frac{9}{4}=\frac{27-32}{16}=-\frac{5}{16}$

$\therefore$ from $\left(1\right)$,

$2\sin \frac{A}{2}\sin \frac{5A}{2}=\frac{1}{8}+\frac{5}{16}=\frac{7}{16}$

$\therefore 32\sin \frac{A}{2}\sin \frac{5A}{2}=16\times \frac{7}{16}=7$ Ans.