Question

If $\tan \frac{\pi }{24}=(\sqrt{p}-\sqrt{q})(\sqrt{r}-s)$, then the value of $p+q+r+s$ is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (C)

Using $\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$
$\tan 7.5^{\circ }=\frac{\sin 15^{\circ }}{1+\cos 15^{\circ }}=\frac{\sin 15^{\circ }}{1+\cos 15^{\circ }}\times \frac{1-\cos 15^{\circ }}{1-\cos 15^{\circ }}$
$=\frac{\sin 15^{\circ }\left(1-\cos 15^{\circ }\right)}{{\sin }^{2}15^{\circ }}=\frac{1-\cos 15^{\circ }}{\sin 15^{\circ }}$
$=\frac{1-\frac{\sqrt{3}+1}{2\sqrt{2}}}{\frac{\sqrt{3}-1}{2\sqrt{2}}}=\frac{2\sqrt{2}-\sqrt{3}-1}{\sqrt{3}-1}$
$=\frac{2\sqrt{2}-\sqrt{3}-1}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}$
$=(\sqrt{3}-\sqrt{2})(\sqrt{2}-1)$