Question

Simplify the expression.

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

Short answer

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

Step-by-step solution

Simplify the expression

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

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

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

Simplify further

Simplify the expression as shown below:

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

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