Question

Trigonometric Substitution Make the indicated trigonometric

substitution in the given algebraic expression and simplify. Assume that $\mathbf{0}\mathbf{\text{<}}\mathit{\theta }\mathbf{\text{<}}\frac{\mathbf{\pi }}{\mathbf{2}}$

$\frac{\mathbf{x}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$

Short answer

The simplified expression is$\frac{\mathbf{x}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}}\mathbf{=}\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }$

Step-by-step solution

Step 1. Given information

An expression $\frac{x}{\sqrt{1-x^2}}$ and value for substitution as$x=\sin \theta$

Step 2. Concept used

Here substitute the value of$x$ as mentioned in the question. After that, simplify it by using trigonometric identities. The simplified outcome is the answer.

Step 3. Calculation

Now, substitute the value in $\frac{x}{\sqrt{1-x^2}}$and simplify:

$\begin{array}{l}\frac{x}{\sqrt{1-x^2}}=\frac{\sin \theta }{\sqrt{1-{\sin }^2\theta }}\\ \because {\sin }^2\theta +{\cos }^2\theta =1\\ =\frac{\sin \theta }{\cos \theta }\\ =\tan \theta \end{array}$

After simplification, we get-

$\tan \theta$