The value of ^-1(√1+ ^2(2)-1 (2))- ^-1(√1+ ^2((1)/(2))+1 ((1)/(2))) is equal to

Question

The value of ${\cot }^{-1}(\frac{\sqrt{1+{	an }^{2}(2)}-1}{	an (2)})-{\cot }^{-1}(\frac{\sqrt{1+{	an }^{2}(\frac{1}{2})}+1}{	an (\frac{1}{2})})$ is equal to

Accepted Answer

$\cot ^{-1}\left(\frac{|\sec 2|-1}{\tan 2}\right)-\cot ^{-1}\left(\frac{\left|\sec \frac{1}{2}\right|+1}{\tan \frac{1}{2}}\right)$

$$\begin{aligned}
& =\cot ^{-1}\left(\frac{-1-\cos 2}{\sin 2}\right)-\cot ^{-1}\left(\frac{1+\cos \frac{1}{2}}{\sin \frac{1}{2}}\right) \
& =\pi-\cot ^{-1}(\cot 1)-\cot ^{-1}\left(\cot \frac{1}{4}\right) \
& =\pi-1-\frac{1}{4}=\pi-\frac{5}{4}
\end{aligned}$$

Solutions · 1