Question

Write the following in the simplest form:

`sin{2tan^-1sqrt((1-x)/(1+x))}`

Answer 1

Let x = cos θ

Now,

`sin{2tan^-1sqrt((1-x)/(1+x))}=sin{2tan^-1sqrt((1-costheta)/(1+costheta))}`

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

`=sin{2tan^-1(tan  theta/2)}`

= sin θ

= sin (cos^-1 x)

`=sin(sin^-1(sqrt(1-x^2)))`

`=sqrt(1-x^2)`