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Chapter 5: Problem 42

Verify each identity. $$\frac{\tan 2 \theta+\cot 2 \theta}{\sec 2 \theta}=\csc 2 \theta$$

Short Answer

Expert verified
The given identity \(\frac{\tan 2 \theta+\cot 2 \theta}{\sec 2 \theta}=\csc 2 \theta\) holds true when broken down and simplified through several steps of trigonometric transformations.

Step by step solution

01

Transform all terms into sine and cosine

Transform each trigonometric term into its sine and cosine equivalent. This gives us: \(\frac{\frac{\sin 2 \theta}{\cos 2 \theta}+\frac{\cos 2 \theta}{\sin 2 \theta}}{\frac{1}{\cos 2 \theta}}\)
02

Simplify the expression

Combining like fractions into one and simplifying gives: \( \frac{\sin^2 2 \theta+\cos^2 2 \theta}{\cos 2 \theta*\sin 2 \theta}\). From here we know that \(\sin^2 \theta + \cos^2 \theta = 1\), therefore the term in the numerator simplifies to \( \frac{1}{\cos 2 \theta*\sin 2 \theta}\)
03

Use the reciprocal identities

By trigonometric identities, we know that \(\frac{1}{\sin \theta} = \csc \theta\) and \(\frac{1}{\cos \theta} = \sec \theta\). So, we can transform the above expression into: \( \csc 2 \theta*\sec 2 \theta\)
04

Use the identity \(\sec \theta*\csc \theta = \csc 2 \theta\)

By trigonometric identities, we know that \(\sec \theta*\csc \theta\) is equal to \(\csc 2 \theta\).\ So the expression becomes: \(\csc 2 \theta\). As the output of step 4 matches the right side of the given identity, we have verified the identity

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