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Question

If x < 0, then write the value of cos−11-x21+x2 in terms of tan−1 x.

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Solution

Let x=tany
Then,
cos-11-x21+x2=cos-11-tan2y1+tan2y=cos-1cos2y 1-tan2x1+tan2x=cos2x=2y ...1

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

-a=tanyy=tan-1-a
Now,

cos-11-x21+x2=2y Using 1=2tan-1-a =-2tan-1x x=-a

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