Find the principal value of the following :\(\tan^{-1}1+\cos^{-1}\le

1.	Find the principal value of the following :\(\tan^{-1}1+\cos^{-1}\left(-\frac{1}{2}\right)+ \sin^{-1}\left(-\frac{1}{2}\right)\)
Answer» [Formula: cos ^-1(-x)=π – cos(x) and sin ^-1(-x)= -sin(x) ] \(\tan^{-1}1+\left(\pi-\cos^{-1}\left(\frac{1}{2}\right)\right)+\left(-\sin^{-1}\left(\frac{1}{2}\right)\right)\) Putting the values for each of the inverse trigonometric terms \(=\frac{\pi}{4}+\left(\pi-\frac{\pi}{3}\right)-\frac{\pi}{6}\) \(=\frac{\pi}{12}+\frac{2\pi}{3}\) \(=\frac{9\pi}{12}=\frac{3\pi}{4}\)

Find the principal value of the following :\(\tan^{-1}1+\cos^{-1}\left(-\frac{1}{2}\right)+ \sin^{-1}\left(-\frac{1}{2}\right)\)

Answer»

[Formula: cos ^-1(-x)=π – cos(x) and sin ^-1(-x)= -sin(x) ]

\(\tan^{-1}1+\left(\pi-\cos^{-1}\left(\frac{1}{2}\right)\right)+\left(-\sin^{-1}\left(\frac{1}{2}\right)\right)\)

Putting the values for each of the inverse trigonometric terms

\(=\frac{\pi}{4}+\left(\pi-\frac{\pi}{3}\right)-\frac{\pi}{6}\)

\(=\frac{\pi}{12}+\frac{2\pi}{3}\)

\(=\frac{9\pi}{12}=\frac{3\pi}{4}\)