Question

The value of \( \tan\left(\frac{\pi}{4} + \theta\right) \times \tan\left(\frac{3\pi}{4} + \theta\right) \) is:

• A. \( -2 \)
• B. \( 2 \)
• C. \( 1 \)
• D. \( -1 \)

Hint:

For products of tangents of sum angles, use the tangent sum formula to simplify the expression.

Answer 1

The Correct Option is D

Step 1: Use the sum of angles formula for tangent. We know that: \[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan(\theta)}{1 - \tan(\theta)}, \] and \[ \tan\left(\frac{3\pi}{4} + \theta\right) = \frac{-1 + \tan(\theta)}{1 + \tan(\theta)}. \] Step 2: Multiply the two expressions. Now, multiply the two tangents: \[ \tan\left(\frac{\pi}{4} + \theta\right) \times \tan\left(\frac{3\pi}{4} + \theta\right) = \left( \frac{1 + \tan(\theta)}{1 - \tan(\theta)} \right) \times \left( \frac{-1 + \tan(\theta)}{1 + \tan(\theta)} \right). \] Simplifying the product, we get: \[ = \frac{(1 + \tan(\theta))(-1 + \tan(\theta))}{(1 - \tan(\theta))(1 + \tan(\theta))} = -1. \] Thus, the correct answer is: \[ \boxed{-1}. \]