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 - Summary Exercises on Verifying Trigonometric Identities - Page 245: 2


$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$ The equation is an identity as proved below.

Work Step by Step

$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$ Take a look at the left side. $$X=\csc\theta\cos^2\theta+\sin\theta$$ - Reciprocal identity: $$\csc\theta=\frac{1}{\sin\theta}$$ Therefore, $$X=\frac{1}{\sin\theta}\times\cos^2\theta+\sin\theta$$ $$X=\frac{\cos^2\theta}{\sin\theta}+\sin\theta$$ $$X=\frac{\cos^2\theta+\sin^2\theta}{\sin\theta}$$ - Pythagorean identity: $\cos^2\theta+\sin^2\theta=1$ So, $$X=\frac{1}{\sin\theta}$$ Again, the first reciprocal identity would be applied here to rewrite $X$ into $\csc\theta$. $$X=\csc\theta$$ In conclusion, $$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$ The equation has been verified to be an identity.
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