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Chapter 5: Problem 42

Verify the identity. $$\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\frac{1-\cos \theta}{|\sin \theta|}$$

Short Answer

Expert verified
The identity \(\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=\frac{1-\cos \theta}{|\sin\theta|}\) is verified to be true after applying the Pythagorean identity and simplifying the expression under the square root.

Step by step solution

01

Express the denominators in terms of \(\sin \theta\)

Use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\). Rearrange this equation to express each \(\cos \theta\) in the denominators in terms of \(\sin \theta\). The right side of the expression becomes: \[\sqrt{\frac{1-\sqrt{1-\sin^2\theta}}{1+\sqrt{1-\sin^2\theta}}}\]
02

Simplify the expression under the square root

Combine the fractions under the square root and simplify the expression. The equation becomes \(\sqrt{\frac{2-\sin^2\theta}{2+\sin^2\theta}}\)
03

Remove the square root

Simplify the expression under the square root by using the property that \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) where a > 0 and b > 0. Thus, \[\frac{\sqrt{2-\sin^2\theta}}{\sqrt{2+\sin^2\theta}}\]
04

Use the absolute value properties of sine

Remember that \(\sqrt{a^2}=|a|\). Replace \(\sqrt{2-\sin^2\theta}\) by \(|\sin\theta|\) and simplify the expression to get the final result, \[\frac{1-\cos \theta}{|\sin\theta|}\]

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