For the principal values, evaluate the following: sec^-1(√2) + 2 cosec^-1(-√2)

Question

For the principal values, evaluate the following:

\(sec^{-1}(\sqrt2)+2cosec^{-1}(-\sqrt2)\)

Answer 1

The range of principal value of sec^–1is [0, π]–{ \(\frac{\pi}2\)}

and sec\((\frac{3\pi}4)=-\sqrt2.\)

The range of principal value of cosec^–1 is

\([\frac{-\pi}2,\frac{\pi}2]\) - {0} and cosec\((\frac{-\pi}4)=-\sqrt2\)

Therefore, the principal value of cosec^–1(–√2) is \(\frac{-\pi}4.\) 

The range of principal value of cosec^–1 is

\([\frac{-\pi}2,\frac{\pi}2] \)- {0} and cosec\((\frac{-\pi}4)=-\sqrt2\)

Therefore, the principal value of cosec^–1(–√2) is \(\frac{-\pi}4.\)