If `x lt 0`, then prove that `cos^(-1) x = pi - sin^(-1) sqrt(1 - x^(2))`

Explanation

Let `x = cos theta`
Since `x lt 0`, we have `theta in [pi//2, pi]`
Now, `tan^(-1). (sqrt(1 - x^(2))/(x)) = tan^(-1). (sqrt(1 - cos^(2) theta)/(cos theta))`
`= tan^(-1) (tan theta) != [ :' theta in (-pi//2, pi//2)]`
`=tan^(-1) tan (pi + (theta - pi)`
`= theta - pi`
`rArr cos^(-1) x = pi + tan^(-1). (sqrt1 - x^(2))/(x)`