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  4. The sum √(5/4)+√(3/2)+√(5/4)-√(3/2) is equal to

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

Continuity and Differentiability

Solution:

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 $S=\sqrt{\frac{5}{4}+\frac{\sqrt{24}}{4}}+\sqrt{\frac{5}{4}-\frac{\sqrt{24}}{4}}=\frac{\sqrt{5+2 \sqrt{6}}+\sqrt{5-2 \sqrt{6}}}{2}=\frac{(\sqrt{3}+\sqrt{2})+(\sqrt{3}-\sqrt{2})}{2}=\sqrt{3}$