The value of [1 2 ^ -1 (2 3 ) ]: - Vidyadip

MCQ

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

✓
$ \frac { 1 }{ \sqrt { 5 } }$
B
$ \frac { 1 }{ \sqrt { 7 } }$
C
$ \frac { 1 }{ \sqrt { 8 } }$
D
$ \frac { 1 }{ \sqrt { 9 } }$

✓

Answer

Correct option: A.

$ \frac { 1 }{ \sqrt { 5 } }$

We have, $\tan\bigg[\frac{1}{2}\cos^{-1}\Big(\frac{2}{3}\Big)\bigg]$
Put $ \cos ^{ -1 }{ \frac { 2 }{ 3 } } =θ$
$ \Rightarrow \cos { \theta } =\frac { 2 }{ 3 }$
$\therefore\tan\bigg[\frac{1}{2}\cos^{-1}\Big(\frac{2}{3}\Big)\bigg]$
$ \therefore \ \tan\frac{\theta}{2}=\sqrt {\frac {1-\cos {\theta}}{1+\cos{\theta }}}$
$=\sqrt {\frac {1-\frac{2}{3} }{1+{\frac{ 2}{3}}}}$
$=\sqrt { \frac { \frac{ 1 }{ 3 } }{ \frac{5}{3} } } $
$=\frac { 1 }{ \sqrt { 5 } }$

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