The value of tan [ 1 2 cos - 1 ( 5 3 ) ] is:

Question

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

Smart Achievers · Accepted Answer

(2)

Ans. $\frac{3 - \sqrt{5}}{2}$

Explanation:

Let $y = \tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$

Putting, $x = \cos^{- 1} (\frac{\sqrt{5}}{3})$

$\Rightarrow$ $\cos x = \frac{\sqrt{5}}{3}$ ...(i)

Now, $y = \tan (\frac{x}{2})$

$\Rightarrow$ $y = \sqrt{\frac{1 - \cos x}{1 + \cos x}}$

$\Rightarrow$ $y = \sqrt{\frac{1 - \frac{\sqrt{5}}{3}}{1 + \frac{\sqrt{5}}{3}}}$ [From (i)]

$\Rightarrow$ $y = \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}}$ $= \sqrt{\frac{(3 - \sqrt{5}) (3 - \sqrt{5})}{(3 + \sqrt{5}) (3 - \sqrt{5})}}$

$\Rightarrow$ $y = \sqrt{\frac{{(3 - \sqrt{5})}^{2}}{9 - 5}}$ $= \frac{3 - \sqrt{5}}{\sqrt{4}}$

$\Rightarrow$ $y = \frac{3 - \sqrt{5}}{2}$

i.e., $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})] = \frac{3 - \sqrt{5}}{2}$

Want Us to Email you About Special Offers & Updates?