Question

The sum $\sqrt{\frac{5}{4}+\sqrt{\frac{3}{2}}}+\sqrt{\frac{5}{4}-\sqrt{\frac{3}{2}}}$ is equal to

• A. $\tan \frac{\pi }{3}$
• B. $\cot \frac{\pi }{3}$
• C. $\sec \frac{\pi }{3}$
• D. $\sin \frac{\pi }{3}$

Answer 1

Answer: (B)

Let $x=\sqrt{\frac{5}{4}+\sqrt{\frac{3}{2}}}+\sqrt{\frac{5}{4}-\sqrt{\frac{3}{2}}}\Rightarrow x^2=\frac{5}{2}+2\sqrt{\frac{25}{16}-\frac{3}{2}}=\frac{5}{2}+2\cdot \frac{1}{4}=3$
$\Rightarrow x=\sqrt{3}=\tan \frac{\pi }{3}$.
Alternative :
Let $S=\sqrt{\frac{5}{4}+\frac{\sqrt{24}}{4}}+\sqrt{\frac{5}{4}-\frac{\sqrt{24}}{4}}=\frac{\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}}}{2}=\frac{(\sqrt{3}+\sqrt{2})+(\sqrt{3}-\sqrt{2})}{2}=\sqrt{3}$