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Exercises Trigonometry

Example 7 from Algebra and Trigonometry, 9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

Verifying an Identity Involving Cosines and Cotangents

Verify the identity: $(1 - \cos^{2} x) (1 + \cot^{2} x) = 1.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We will work on the left side of the equation.

$\begin{matrix} (1 - \cos^{2} x) (1 + \cot^{2} x) & = & (1 - \cos^{2} x) (1 + \frac{\cos^{2} x}{\sin^{2} x}) \\ = & (1 - \cos^{2} x) (\frac{\sin^{2} x}{\sin^{2} x} + \frac{\cos^{2} x}{\sin^{2} x}) & Find the common denominator . \\ = & (1 - \cos^{2} x) (\frac{\sin^{2} x + \cos^{2} x}{\sin^{2} x}) \\ = & (\sin^{2} x) (\frac{1}{\sin^{2} x}) \\ = & 1 \end{matrix}$

How did it go?

PreviousVerifying an Identity Using Algebra and Even/Odd Identities Next Writing the Trigonometric Expression as an Algebraic Expression

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.