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Question
Write the following in the simplest form:
`tan^-1sqrt((a-x)/(a+x)),-a<x<a`
Solution
Let `x = acos theta`
Now,
`tan^-1sqrt((a-x)/(a+x))=tan^-1sqrt((a-acostheta)/(a+acostheta))`
`=tan^-1sqrt((1-costheta)/(1+costheta))`
`=tan^-1sqrt((2sin^2 theta/2)/(2cos^2 theta/2))`
`=tan^-1(tan theta/2)`
`=theta/2`
`=1/2cos^-1(x/a)`
`thereforetan^-1sqrt((a-x)/(a+x))=(cos^-1(x/a))/2`
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