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

Answer

$\cos^4{x}-\sin^4{x}=\cos{(2x)}$

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, $$\cos^4{x}-\sin^4{x}=\cos{(2x)}$$
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