The value of is \cot \left (cosec^{-1}\frac{5}{3} +\tan ^{-1}\frac{2}{3}\right )

  • Option 1)

    \frac{5}{17}

  • Option 2)

    \frac{6}{17}

  • Option 3)

    \frac{3}{17}

  • Option 4)

    \frac{4}{17}

 

Answers (1)

As we learnt in 

Results of Compound Angles -

\cot \left ( A+B \right )= \frac{\cot A\cot B-1}{\cot A+\cot B}

- wherein

Where A and B are two angles.

 

 \cot \left (cosec^{-1}\frac{5}{3} +tan^{-1}\frac{2}{3} \right ) = \frac{cot\left ( cosec^{-1}\frac{5}{3} \right )\cot \left ( tan^{-1} \frac{2}{3}\right )-1} {cot \:cosec ^{-1}\frac{5}{3}+\cot \: \tan ^{-1}\frac{2}{3}}

Now    \cot cosec^{-1} \frac{5}{3}=\frac{4}{3}                \left [ cosec \theta = \frac{5}{3} \Rightarrow sin\theta=\frac{3}{5},\ cos\theta=\frac{4}{5},\ cot\theta=\frac{4}{3} \right ]

Thus expression becomes 

\frac{\frac{4}{3}\times \frac{3}{2}-1}{\frac{4}{3}+\frac{3}{2}}                                \left[\cot tan^{-1}\frac{2}{3}=\frac{2}{3} \right ]

=\frac{6}{17}


Option 1)

\frac{5}{17}

Incorrect

Option 2)

\frac{6}{17}

Correct

Option 3)

\frac{3}{17}

Incorrect

Option 4)

\frac{4}{17}

Incorrect

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