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