Question

If $\cos A=\frac{3}{4}$, then the value of $16 \cos ^2\left(\frac{A}{2}\right)-32 \sin \left(\frac{A}{2}\right) \sin \left(\frac{5 A}{2}\right)$ is

• A. $-4$
• B. $-3$
• C. 3
• D. 4

Answer 1

Answer: (C)

$\cos A=\frac{3}{4}$
$16 \cos ^2 \frac{A}{2}-32 \sin \frac{A}{2} \sin \frac{5 A}{2}$
$=\frac{16(1+\cos A)}{2}-16(\cos 2 A-\cos 3 A)$
$=\frac{16(1+\cos A)}{2}-16\left\{\left(2 \cos ^2 A-1\right)-\left(4 \cos ^3 A-3 \cos A\right)\right\}$
$=8\left(1+\frac{3}{4}\right)-16\left\{2 \times \frac{9}{16}-1-4 \times \frac{27}{64}+3 \times \frac{3}{4}\right\}=3$