Question

Write the following in the simplest form:

`tan^-1sqrt((a-x)/(a+x)),-a<x<a`

Answer 1

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`