# 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) $\frac{4}{17}$

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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