NIMCET 2018 — Mathematics PYQ

Concept:
$$\cot (\theta +\varphi )=\frac{(\cot \varphi )(\cot \theta )-1}{(\cot \theta )+(\cot \varphi )}$$

Calculations:
Consider ${\mathrm{cosec}}^{-1}\frac{5}{3}=\theta$ and ${\tan }^{-1}\frac{2}{3}=\varphi$
$\Rightarrow \mathrm{cosec}\theta =\frac{5}{3}$ and $\tan \varphi =\frac{2}{3}$

From Figure, we have
$$\cot \theta =\frac{4}{3}\text{ and }\cot \varphi =\frac{3}{2}$$
Now, consider $\cot \left({\mathrm{cosec}}^{-1}\frac{5}{3}+{\tan }^{-1}\frac{2}{3}\right)$
$$<br>\cot (\theta +\varphi )=\frac{(\cot \varphi )(\cot \theta )-1}{(\cot \theta )+(\cot \varphi )}$$
$$\cot (\theta +\varphi )=\frac{\left(\frac{4}{3}\right)\left(\frac{3}{2}\right)-1}{\left(\frac{4}{3}\right)+\left(\frac{3}{2}\right)}$$
$$<br>(37)$$
$$\cot (\theta +\varphi )=\frac{6}{17}$$
Hence, The value of $\cot \left({\mathrm{cosec}}^{-1}\frac{5}{3}+{\tan }^{-1}\frac{2}{3}\right)$ is $6/17$