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