Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Trigonometry

Example 10 from Algebra and Trigonometry, 9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

\csc^{2} θ - \cot^{2} θ

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 can start with the Pythagorean identity.

$1 + \cot^{2} θ = \csc^{2} θ$

Now we can simplify by substituting $1 + \cot^{2} θ$ for $\csc^{2} θ .$ We have

$\begin{matrix} \csc^{2} θ - \cot^{2} θ & = & 1 + \cot^{2} θ - \cot^{2} θ \\ = & 1 \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use algebraic techniques to verify the identity: $\frac{\cos θ}{1 + \sin θ} = \frac{1 - \sin θ}{\cos θ} .$

(Hint: Multiply the numerator and denominator on the left side by $1 - \sin θ .)$

How did it go?

PreviousRewriting a Trigonometric Expression Using the Difference of Squares Next Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

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.