Question

If $cosA=3/4$, then the value of $987\left[16co{s}^2\right(A/2)-32sin(A/2\left)sin\right(5A/2)$ is equal to

Answer 1

Consider ,$987\left[16{\cos }^2\right(A/2)-32\sin (A/2\left)\sin \right(5A/2)$
$=987\times 16\left[{\cos }^2\right(A/2)-(\cos 2A-\cos 3A\left)\right]$ ....(1)
Given $\cos A=\frac{3}{4}$
$\Rightarrow 2{\cos }^2\left(A/2\right)-1=\frac{3}{4}$
$\Rightarrow {\cos }^2\left(A/2\right)=\frac{7}{8}$
Also $\cos 2A=2{\cos }^{2}A-1$
$\Rightarrow \cos 2A=\frac{1}{8}$
Also,$\cos 3A=4{\cos }^{3}A-3\cos A$
$\Rightarrow \cos 3A=-\frac{9}{16}$
$=987\times 16[\frac{7}{8}-\frac{1}{8}-\frac{9}{16}]$
$=987\times 16\times \frac{3}{16}=2961$.