Question

Write the following in the simplest form:

`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`

Answer 1

Let `x = asintheta`

Now,

`tan^-1{x/(a+sqrt(a^2-x^2))}=tan^-1{(asintheta)/(a+sqrt(a^2-a^2cos^2theta))}`

`=tan^-1{(asintheta)/(a+asqrt(cos^2theta))}`

`=tan^-1{sintheta/(1+costheta)}`

`=tan^-1{(2sin(theta/2)cos(theta/2))/(2cos^2  theta/2)}`

`=tan^-1{tan  theta/2}`

`=theta/2`

`=1/2sin^-1(x/a)`