First slide

Properties of ITF

Question

The value of $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$ is

Easy

A

$\frac{3 + \sqrt{5}}{2}$
B

$3 + \sqrt{5}$
C

$\frac{1}{2} (3 - \sqrt{5})$
D

$2 - \sqrt{3}$

Solution

$L e t \cos^{- 1} (\frac{\sqrt{5}}{3}) = x . \Rightarrow \cos x = \frac{\sqrt{5}}{3} ∴ \tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})] = \tan \frac{x}{2} = \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \sqrt{\frac{1 - \frac{\sqrt{5}}{3}}{1 + \frac{\sqrt{5}}{3}}} = \frac{3 - \sqrt{5}}{2}$

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