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A value of x satisfying the equation

$\sin \left [ \cot ^{-1}\left ( 1+ x \right ) \right ] = \cos \left [ \tan ^{-1}x \right ],$ is

Option 1)
$-\frac{1}{2}$

Option 2)
$-1$

Option 3)
$0$

Option 4)
$\frac{1}{2}$

Answers (1)

best_answer

As we learnt in

Trigonometric Equations -

The equations involving trigonometric function of unknown angles are known as trigonometric equations.

- wherein

e.g. $\cos ^{2}\Theta - 4\cos \Theta = 1$

$\sin \cot^{-1}(1+x)=\cos \tan^{-1}x$

Let, $\cot^{-1}(1+x)=a$ and $\tan^{-1}x=b$

$\Rightarrow \cot a=1+x$ and $\tan b=x$

$\Rightarrow sin a=\frac{1}{\sqrt{1+(1+x)^{2}}}$ ..............( 1 )

$\Rightarrow cos b = \frac{1}{\sqrt{1+x^{2}}}$ .............. ( 2 )

From ( 1 ) and ( 2 ),

$\frac{1}{\sqrt{1+(1+x)^{2}}}= \frac{1}{\sqrt{1+x^{2}}}$

$\Rightarrow 1+(1+x)^{2}=1+x^{2}$

$\Rightarrow x^{2}+2x+1=x^{2}$

$\Rightarrow x=-\frac{1}{2}$

Option 1)

$-\frac{1}{2}$

This option is correct.

Option 2)

$-1$

This option is incorrect.

Option 3)

$0$

This option is incorrect.

Option 4)

$\frac{1}{2}$

This option is incorrect.

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