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Chapter 5: Problem 41

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

Short Answer

Expert verified
After using trigonometric identities to simplify the given equation, we find that indeed, \(\frac{\tan 2 \theta+\cot 2 \theta}{\csc 2 \theta} = \sec 2 \theta\).

Step by step solution

01

Expand using Trigonometric Identities

Expand the equations using the basic trigonometric identities. Write \(\tan 2 \theta\) as \(\frac{\sin 2 \theta}{\cos 2 \theta}\), \(\cot 2 \theta\) as \(\frac{\cos 2 \theta}{\sin 2 \theta}\), \(\csc 2 \theta\) as \(\frac{1}{\sin 2 \theta}\), and \(\sec 2 \theta\) as \(\frac{1}{\cos 2 \theta}\). We get: \(\frac{\frac{\sin 2 \theta}{\cos 2 \theta} +\frac{\cos 2 \theta}{\sin 2 \theta}}{\frac{1}{\sin 2 \theta}}\).
02

Simplify the Complex Fraction

Now, let's simplify this complex fraction. To do this, multiply the numerator and the denominator by \(\sin 2 \theta\cos 2 \theta\) (the LCD of the complex fraction). The equation simplifies to: \(\frac{\sin^2 2 \theta+\cos^2 2 \theta}{\sin 2 \theta}\).
03

Apply Pythagorean Identity

The top of the fraction, \(\sin^2 2 \theta+\cos^2 2 \theta\), is the Pythagorean Identity and equals 1, as per the Pythagorean identity. So we are left with \(\frac{1}{\sin 2 \theta}\).
04

Demonstrate Equality

Our final equation \(\frac{1}{\sin 2 \theta}\) is equivalent to our original right hand side \(\sec 2 \theta\), which is \(\frac{1}{\cos 2 \theta}\). Thus, our original equation \(\frac{\tan 2 \theta+\cot 2 \theta}{\csc 2 \theta} = \sec 2 \theta\) has been verified.

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