Question

The maximum value of $\cos x\left\{\sin x+\sqrt{\sin ^{2} x+\sin ^{2} \frac{\pi}{6}}\right\}$ is

• A. $\frac{\sqrt{5}}{3}$
• B. $\frac{\sqrt{5}}{2}$
• C. $\sqrt{\frac{5}{2}}$
• D. $\sqrt{\frac{3}{2}}$

Answer 1

Answer: (B)

Let $y=\cos x\left\{\sin x+\sqrt{\sin ^{2} x+\sin ^{2} \frac{\pi}{6}}\right\}$ is
$\Rightarrow y^{2}\left(1+\tan ^{2} x\right)-2 y \tan x=\frac{1}{4}$
$\Rightarrow y^{2} \tan ^{2} x-2\, y\, \tan x+\left(y^{2}-\frac{1}{4}\right) \geq 0$
For real $\tan x$,
disc. $\geq 0$
$\Rightarrow 4 y^{2}-4 y^{2}\left(y^{2}-\frac{1}{4}\right) \geq 0$
$\Rightarrow 0 \leq y^{2} \leq \frac{5}{4}$
$\therefore$ Maximum value of $y=\frac{\sqrt{5}}{2}$