Question

The greatest and least values of (sin–¹x)² + (cos^–1x)² are respectively ______.

Options

• `(5pi^2)/4` and `pi^2/8`

• `pi/2` and `(-pi)/2`

• `pi^2/4` ad `(-pi^2)/4`

• `pi^2/4` and 0

Answer 1

The greatest and least values of (sin–¹x)² + (cos^–1x)² are respectively `(5pi^2)/4` and `pi^2/8`.

Explanation:

We have (sin^–1x)² + (cos^–1x)²

= (sin^–1x + cos^–1x)² – 2 sin^–1x cos^–1x

=  `pi^2/4 - 2sin^1x (pi/2 - sin^-1x)`

= `pi^2/4 - pi sin^-1x + 2(sin^-1x)^2`

= `2[(sin^-1x)^2 - pi/2 sin^-1x + pi^2/8]`

= `2[(sin^-1x - pi/4)^2 + pi^2/16]`

Thus, the least value is `2(pi^2/16)`

i.e. `pi^2/8` and the Greatest value is `2[((-pi)/2 - pi/4)^2 + pi^2/16]`

i.e. `(5pi^2)/4`.