Question

^{tan-1 √3 -sec-1(-2)+cosec-12/√3}

Answer 1

we know \( \tan(\frac{\pi}{3})=\sqrt{3} \) so \( \tan^{-1} (\sqrt{3})=\frac{\pi}{3} \)

we know \( \cos(\frac{2\pi}{3})=-\frac{1}{2} \) so \( \sec(\frac{2\pi}{3})=\frac{1}{\cos(\frac{2\pi}{3})}=-2 \) and finally \( \sec^{-1} (-2)=\frac{2\pi}{3} \)

and also we know \( \sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2} \) so \( \csc(\frac{\pi}{3})=\frac{1}{\sin(\frac{\pi}{3})}=\frac{2}{\sqrt{3}} \) and finally \( \csc^{-1}(\frac{2}{\sqrt{3}})=\frac{\pi}{3} \)

\( \tan^{-1}(\sqrt{3})-\sec^{-1}(-2)+\csc^{-1}(\frac{2}{\sqrt{3}})=\frac{\pi}{3}-\frac{2\pi}{3}+\frac{\pi}{3}=0 \)