# if then is equal to : Option 1) Option 2) Option 3) Option 4)

D Divya Saini

As we learnt in

Inverse Trigonometric Functions -

The functions $\sin ^{-1}x, \cos ^{-1}x, \tan ^{-1}x, \cot ^{-1}x, \csc ^{-1}x$ and $\sec ^{-1}x$ are the inverse trigonometric functions.

- wherein

If $\sin x = \frac{1}{2}$

then, $x = \sin ^{-1}\frac{1}{2}$

$f(5)=2\tan^{-1}5+\sin^{-1}\frac{2\times 5}{1+25}$$=2\tan^{-1}5+\sin^{-1}\frac{5}{13}$

$=2\tan^{-1}5+\tan^{-1}\frac{5}{12}$            $\left [ \sin^{-1}\frac{5}{13}=\tan^{-1}\frac{5}{12} \right ]$

$=\tan^{-1}\frac{2\times 5}{1-25}+\tan^{-1}\frac{5}{12}$

$=\tan^{-1}\left( -\frac{5}{12}\right)=\tan^{-1}\frac{5}{12}$

now,

$\tan [f(5)]=\tan\: \left[\tan^{-1} \frac{5}{12}+\tan^{-1}\left ( -\frac{5}{12} \right )\right]=0$

$\Rightarrow [f(5)]=\tan^{-1}0=0\:\:or\:\pi$                .......................... ( 1 )

Now,

$x>0,\:\tan^{-1} 5>0,\:\sin\frac{5}{13}>0$             ............................( 2 )

$\Rightarrow f(5)>0$

$\Rightarrow f(5)=\pi$                            [From ( 1 ) and ( 2 )]

Option 1)

This option is incorrect.

Option 2)

This option is correct.

Option 3)

This option is incorrect.

Option 4)

This option is incorrect.

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