Question

If $\alpha ϵ(0,\frac{\pi }{2})$, then the value of ${\tan }^{-1}\left(\cot \alpha \right)-{\cot }^{-1}\left(\tan \alpha \right)+{\sin }^{-1}\left(\sin \alpha \right)-{\cos }^{-1}\left(\cos \alpha \right)$ is equal to

• A. $2\alpha$
• B. $\pi +\alpha$
• C. $0$
• D. $\pi -2\alpha$

Answer 1

The correct option is C $0$

$ta{n}^{-1}\left(cot\alpha \right)-co{t}^{-1}\left(tan\alpha \right)+si{n}^{-1}\left(sin\alpha \right)-co{s}^{-1}\left(cos\alpha \right)=ta{n}^{-1}\left(tan(\frac{\pi }{2}-\alpha )\right)-co{t}^{-1}\left(cot(\frac{\pi }{2}-\alpha )\right)+si{n}^{-1}sin\alpha -co{s}^{-1}cos\alpha =(\frac{\pi }{2}-\alpha )-(\frac{\pi }{2}-\alpha )+\alpha -\alpha =0$