Question

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

$\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{s}\mathit{e}\mathit{c}\mathit{\theta }$

Short answer

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

Step-by-step solution

Step 1. Given information

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

Step 2. Concept used

Substitute the$x$ value provided in the question here. Then, using trigonometric identities, simplify it. The solution is the simple result.

Step 3. Calculation

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

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

After simplification, we get-

$\tan \theta$