Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.5 Double-Angle Identities - 5.5 Exercises - Page 237: 53



Work Step by Step

Use a graphing utility to graph the given expression. (Refer to the graph below,) Notice that the graph is identical with the graph of $\cos{2x}$. This means that $\cos^4{x}-\sin^4{x}=\cos{(2x)}$. The given expression can be written as: $(\cos^2{x})^2-(\sin^2{x})^2$ Factor the expression using the formula $a^2-b^2=(a-b)(a+b)$ where $a=\cos^2{x}$ and $b=\sin^2{x}$ to obtain: $$(\cos^2{x})^2-(\sin^2{x})^2=\left(\cos^2{x}-\sin^2{x}\right)\left(\cos^2{x}+\sin^2{x}\right)$$ Since $\cos^2{x}+\sin^2{x}=1$. then the equation above simplifies to: \begin{align*} (\cos^2{x})^2-(\sin^2{x})^2&=(\cos^2{x}-\sin^2{x})(1)\\ (\cos^2{x})^2-(\sin^2{x})^2&=(\cos^2{x}-\sin^2{x})\\ \end{align*} Recall: $\cos{(2x)}=\cos^2{x} - \sin^2{x}$ Thus, the equation above becomes: $$(\cos^2{x})^2-(\sin^2{x})^2=\cos{(2x)}$$ Therefore, $$\cos^4{x}-\sin^4{x}=\cos{(2x)}$$
Small 1554571077
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.