Problem 43 - Precalculus Essentials

Question

Verify each identity. $$\tan \left(\theta+\frac{\pi}{4}\right)=\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}$$

Step-by-Step Solution

Answer

Yes, the trigonometric identity is correct. You can show that \( \tan(\theta + \frac{\pi}{4}) = \frac{\cos\theta+\sin\theta}{\cos\theta-\sin\theta} \) by using the addition formula for tangent and simplifying.

Step 1: Add the Angle Formula for Tangent

The addition formula for tangent is \( \tan(A+B) = \frac{\sin A}{\cos B} + \frac{\cos A}{\sin B}\). Substitute \( \theta \) for \( A \) and \( \frac{\pi}{4} \) for \( B \) in the addition formula. So, \( \tan(\theta+\frac{\pi}{4})=\frac{\tan\theta+\tan\frac{\pi}{4}}{1-\tan\theta\tan\frac{\pi}{4}} \).

Step 2: Substitute Tangent values

We know that \( \tan\frac{\pi}{4}=1 \). Substitute \( 1 \) into the equation from Step 1: \( =\frac{\tan\theta+1}{1-\tan\theta}\).

Step 3: Convert Tangent to Sine/Cosine

Use the definition of tangent (\( \tan\theta=\frac{\sin\theta}{\cos\theta} \)). Substitute into the equation from Step 2: \( =\frac{\frac{\sin\theta}{\cos\theta}+1}{1-\frac{\sin\theta}{\cos\theta}} \)

Step 4: Simplify

To simplify the equation, multiply the numerator and the denominator by \( \cos\theta \): \( =\frac{\sin\theta+\cos\theta}{\cos\theta-\sin\theta} \). This gives us the right side of the original equation, confirming the identity is correct.