First slide

Properties of ITF

Question

The value of the expression $\tan (\frac{1}{2} \cos^{- 1} \frac{2}{\sqrt{5}})$ is

Moderate

A

$2 + \sqrt{5}$
B

$\sqrt{5} - 2$
C

$\frac{\sqrt{5} + 2}{2}$
D

$5 + \sqrt{2}$

Solution

$\begin{matrix} \tan (\frac{1}{2} \cos^{- 1} \frac{2}{\sqrt{5}}) = \sqrt{\frac{1 - \cos [\cos^{- 1} (\frac{2}{\sqrt{5}})]}{1 + \cos [\cos^{- 1} (\frac{2}{\sqrt{5}})]}} \\ = \sqrt{\frac{1 - \frac{2}{\sqrt{5}}}{1 + \frac{2}{\sqrt{5}}}} = \sqrt{\frac{\sqrt{5} - 2}{\sqrt{5} + 2} \times \frac{\sqrt{5} - 2}{\sqrt{5} - 2}} \\ = \sqrt{\frac{(\sqrt{5} - 2)^{2}}{5 - 4}} = \sqrt{5} - 2 \end{matrix}$

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