Question

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

• A. -4
• B. -3
• C. 3
• D. 4

Answer 1

Answer: (C)

3

$\begin{array}{c}Wehave\\ \cos A=\frac{3}{4}\\ 16{\cos }^2\frac{A}{2}-32\sin \left(\frac{A}{2}\right)\sin \left(\frac{5A}{2}\right)\\ =8\left(1+\cos A\right)-16\left[\cos 2A-\cos 3A\right]\\ =8\left[1+\frac{3}{4}\right]-16\left[\cos 2A-\cos 3A\right]\\ Now,\\ \cos 2A=2co{s}^{2}A-1\\ =2\times \frac{9}{16}-1=\frac{1}{8}\\ \cos 3A=4{\cos }^{3}A-3\cos A\\ =4\times \frac{27}{64}-\frac{9}{4}\\ =\frac{27-36}{16}=\frac{-9}{16}\\ =8\left(\frac{7}{4}\right)-16\left(\frac{1}{8}+\frac{9}{16}\right)\\ =14-2-9\\ =3\\ Hence,theoptionCiscorrectanswer.\end{array}$