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)

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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