Question

Find domain & range of y = sec^-1 x + cosec^-1 x + tan^-1 x. 

Answer 1

y = sec^-1x + cosec^-1x + tan^-1x

Domain of sec^-1x is R - (2n + 1)\(\frac{\pi}2\); n \(\in\) Z

Domain of cosec^-1x is R  - nπ, n \(\in\) Z

Domain of tan^-1x is R

\(\therefore\)Domain of y is R - {(2n + 1) \(\frac{\pi}2\), nπ}, n \(\in\) Z

\(\because\) sec^-1x + cosec^-1x = \(\frac{\pi}2\)

\(\therefore\) y = \(\frac{\pi}2\) + tan^-1x

Principal Range of tan^-1x is (\(-\frac{\pi}2,\frac{\pi}2\))

\(\therefore\) Range of y is \(\frac{\pi}2+(-\frac{\pi}2,\frac{\pi}2)\) or (0, π).