Question

If x < 0, then write the value of cos⁻¹ `((1-x^2)/(1+x^2))` in terms of tan⁻¹ x.

Answer 1

Let x = tan y

Then,

`cos^-1((1-x^2)/(1+x^2))=cos^-1((1-tan^2y)/(1+tan^2y))`

`=cos^-1(cos2y)`    `[because (1-tan^2x)/(1+tan^2)=cos2x]`

= 2y                 ...(1)

The value of x is negative.
So, let x = -a where a > 0.

`-a = tan y`

`=>y=tan^-1(-a)`

Now,

`cos^-1((1-x^2)/(1+x^2))=2y`         [Using (1)]

`=2tan^-1(-a)`

`=-2tan^-1x`            `[becausex=-a]`