Trigonometry (11th Edition) Clone

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

Chapter 5 - Quiz (Sections 5.1-5.4) - Page 230: 9


$$\frac{\sin^2\theta-\cos^2\theta}{\sin^4\theta-\cos^4\theta}=1$$ The equation is an identity as 2 sides are equal to each other.

Work Step by Step

$$\frac{\sin^2\theta-\cos^2\theta}{\sin^4\theta-\cos^4\theta}=1$$ Let's consider the left side. $$A=\frac{\sin^2\theta-\cos^2\theta}{\sin^4\theta-\cos^4\theta}$$ We see that both numerator and denominator can be expanded using the expansion $X^2-Y^2=(X-Y)(X+Y)$ and $X^4-Y^4=(X^2-Y^2)(X^2+Y^2)$ However, it might be wiser to only expand the denominator now, as you will see later. $$A=\frac{\sin^2\theta-\cos^2\theta}{(\sin^2\theta-\cos^2\theta)(\sin^2\theta+\cos^2\theta)}$$ $$A=\frac{1}{\sin^2\theta+\cos^2\theta}$$ So by only expanding the denominator, the whole numerator has been simplified. Here, we apply Pythagorean Identity: $\sin^2\theta+\cos^2\theta=1$. $$A=\frac{1}{1}=1$$ Thus, both sides are equal to each other, meaning the equation is an identity.
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