Question

Find the value of $\sin \left(2{\cot }^{-1}\left(\frac{5}{12}\right)\right)$

Answer 1

$\sin \left(2{\cot }^{-1}\left(\frac{5}{12}\right)\right)$

Let $\theta ={\cot }^{-1}\left(\frac{5}{12}\right)$

$\Rightarrow \cot \theta =\frac{5}{12}$

$\Rightarrow \tan \theta =\frac{12}{5}$

$\Rightarrow A{C}^2=A{B}^2+B{C}^2$

$={\left(12\right)}^2+{\left(5\right)}^2$

$\Rightarrow AC=\sqrt{169}=13$

Now, $\sin 2\theta =2\sin \theta \cos \theta$

$=2\times \frac{12}{13}\times \frac{5}{13}$

$=\frac{120}{169}.$

Hence, the answer is $\frac{120}{169}.$