First slide

Properties of ITF

Question

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

Moderate

A

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

$3 + \sqrt{5}$
C

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

none of these

Solution

Let $\cos^{- 1} (\frac{\sqrt{5}}{3}) = α .$ Then,
$\cos α = \frac{\sqrt{5}}{3},$ Where $0 < α < \frac{π}{2}$
Now,
$\begin{aligned} \tan \frac{α}{2} = \sqrt{\frac{1 - \cos \bar{α}}{1 + \cos α}} \\ \Rightarrow \tan \frac{α}{2} = \sqrt{\frac{1 - \sqrt{5} / 3}{1 + \sqrt{5} / 3}} \end{aligned}$
$= \tan \frac{α}{2} = \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}} = \sqrt{\frac{(3 - \sqrt{5})^{2}}{9 - 5}} = \frac{1}{2} (3 - \sqrt{5})$
$\Rightarrow \tan \{\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})\} = \frac{3 - \sqrt{5}}{2}$

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