Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 5 - Section 5.1 - Proving Identities - 5.1 Problem Set - Page 280: 64


$$\frac{\csc^2 y+\cot^2 y}{\csc^4 y-\cot^4 y}=1$$ The identity is proved to be true.

Work Step by Step

$$\frac{\csc^2 y+\cot^2 y}{\csc^4 y-\cot^4 y}=1$$ First, factoring the denominator on the left side gives $$\frac{\csc^2 y+\cot^2 y}{\csc^4 y-\cot^4 y}$$ $$=\frac{\csc^2 y+\cot^2 y}{(\csc^2 y-\cot^2 y)(\csc^2 y+\cot^2 y)}$$ $$=\frac{1}{\csc^2 y-\cot^2 y}.$$ Now, remember that $\csc^2 y=1+\cot^2 y$, which we would replace as follows $$=\frac{1}{1+\cot^2 y-\cot^2 y}$$ $$=\frac{1}{1}$$ $$=1$$ Hence the identity is proven.
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