Question

Simplify the expression.

76. \(\tan \left( {{{\sin }^{ - 1}}x} \right)\)

Short answer

The solution of the expression is \(\frac{x}{{\sqrt {1 - {x^2}} }}\).

Step-by-step solution

Simplify the expression

Consider \(y = {\sin ^{ - 1}}x\).

Draw the right triangle with angle \(y\) as shown below:

It is observed from the figure of the right triangle \(\sin y = x\) (which describes the case \(y > 0\)).

Simply further

Simplify the expression as shown below:

\(\begin{aligned}\tan \left( {{{\sin }^{ - 1}}x} \right) &= \tan y\\ &= \frac{x}{{\sqrt {1 - {x^2}} }}\end{aligned}\)

Thus, the solution of the expression is \(\frac{x}{{\sqrt {1 - {x^2}} }}\).