Question

sin (tan^–1 x), |x| < 1 is equal to ______.

Options

• `x/(sqrt(1 - x^2))`

• `1/sqrt(1 - x^2)`

• `1/sqrt(1 + x^2)`

• `x/(sqrt(1 + x^2))`

Answer 1

sin (tan^–1 x), |x| < 1 is equal to `bbunderline (x/(sqrt(1+ x^2)))`.

Explanation:

Let tan^–1 x = y.

Then, tan y = x

⇒ `sin y = x/sqrt(1 + x^2)`

∴ `y = sin^-1 (x/sqrt(1 + x^2))`

⇒ `tan^-1x = sin^-1 (x/sqrt(1 + x^2))`

∴ `sin (tan^-1x) = sin (sin^-1  x/sqrt(1 + x^2))`

= `x/sqrt(1 + x^2)`