The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+ an ^{-1} \frac{2}{3} ight)$ is

Question

The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+	an ^{-1} \frac{2}{3}ight)$ is

Aicharya · Accepted Answer

$\begin{aligned} & \cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+	an ^{-1} \frac{2}{3}ight) \ & =\cot \left(\sin ^{-1} \frac{3}{5}+	an ^{-1} \frac{2}{3}ight) \ & =\cot \left(	an ^{-1} \frac{\frac{3}{5}}{\sqrt{1-\left(\frac{3}{5}ight)^2}}+	an ^{-1} \frac{2}{3}ight) \ & =\cot \left(	an ^{-1} \frac{3}{4}+	an ^{-1} \frac{2}{3}ight) \ & =\cot \left[	an ^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4} 	imes \frac{2}{3}}ight)ight]\end{aligned}$ $\begin{aligned} & =\cot \left[	an ^{-1}\left(\frac{17}{6}ight)ight] \ & =\cot \left[\cot ^{-1}\left(\frac{6}{17}ight)ight]=\frac{6}{17}\end{aligned}$

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

Solution