Question

12.-111-ry = sin

Answer 1

The given equation is y= sin −1 ( 1− x 2 1+ x 2 )

y= sin −1 ( 1− x 2 1+ x 2 )(1)

Let x=tanθ θ= tan −1 x

Hence, y= sin −1 ( 1− tan 2 θ 1+ tan 2 θ ) (2)

Also, cos( 2θ )= 1− tan 2 θ 1+ tan 2 θ (3)

From (2) and (3),

y= sin −1 ( 1− tan 2 θ 1+ tan 2 θ ) = sin −1 ( cos( 2θ ) ) = sin −1 ( sin( π 2 −2θ ) ) = π 2 −2θ

So, y= π 2 −2 tan −1 x

Differentiate both sides,

dy dx =0− 2 1+ x 2

Thus, the derivative of sin −1 ( 1− x 2 1+ x 2 ) is − 2 1+ x 2 .