# Use the trigonometric substitution to write the algebraic expression as a trigonometric function...

## Question:

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 0, where 0 < θ < π/2.

{eq}\sqrt{x^2+100}, \quad x = 10 \tan \theta {/eq}

## Basic Trigonometric Identities:

In trigonometry, we have several basic identities that help us to solve various problems related to six trigonometric functions. Out of these basic identities, Pythagorean identities are the three identities that are based on the Pythagoras theorem and play an important role while solving trigonometric problems.

\begin{align} \sin^2x+\cos^2 x &=1\\[0.3cm] 1+\tan^2 x &=\sec^2 x\\[0.3cm] 1+\cot^2 x &=\csc^2 x \end{align}

Given data:

$$\sqrt{x^2+100}, \quad x = 10 \tan \theta$$

Using the given trigonometric substitution to write the algebraic expression as a trigonometric function of {eq}\theta {/eq}

The trigonometry indentity that we will apply in the given expression is:

$$1+\tan^2 \theta=\sec^2 \theta$$

Substitute the given trigonometric substitution in the given expression:

\begin{align} \sqrt{x^2+100} &=\sqrt{(10\tan \theta)^2+100} \ && \left [ \text{Given}: x = 10 \tan \theta \right ]\\[0.3cm] &=\sqrt{100\tan^2 \theta+100}\\[0.3cm] &=\sqrt{100(1+\tan^2 \theta)}\\[0.3cm] &=\sqrt{100\sec^2 \theta} \ && \left [ \text{Identity}: 1+\tan^2 \theta =\sec^2 \theta \right ]\\[0.3cm] &=\sqrt{(10\sec \theta)^2}\\[0.3cm] &=\boxed{10\sec \theta} \end{align}