Question

The value of $\cos {66}^{\circ }+\sin {84}^{\circ }$ is

• A. $\frac{\sqrt{15}-\sqrt{3}}{4}$
• B. $\frac{\sqrt{15}-3}{4}$
• C. $\frac{\sqrt{15}+\sqrt{3}}{4}$
• D. $\frac{\sqrt{15}+3}{4}$

Answer 1

The correct option is C $\frac{\sqrt{15}+\sqrt{3}}{4}$

$\cos {66}^{\circ }+\sin {84}^{\circ }=\cos {66}^{\circ }+\sin (90-6{)}^{\circ }$

$=\cos {66}^{\circ }+\cos 6^{\circ }$ [because $sin(90-\theta )=cos\theta$]
$=2\cos \frac{60}{2}\cos \frac{72}{2}$
$=2\cos {30}^o\cos {36}^o$

$=2\times \frac{\sqrt{3}}{2}\times \left(\frac{\sqrt{5}+1}{4}\right)$
$=\frac{\sqrt{15}+\sqrt{3}}{4}$
Hence, option 'C' is correct.