Question:
The value of \( \tan\left(\frac{\pi}{4} + \theta\right) \times \tan\left(\frac{3\pi}{4} + \theta\right) \) is:
The value of \( \tan\left(\frac{\pi}{4} + \theta\right) \times \tan\left(\frac{3\pi}{4} + \theta\right) \) is:
Show Hint
For products of tangents of sum angles, use the tangent sum formula to simplify the expression.
Updated On: Oct 18, 2025
- \( -2 \)
- \( 2 \)
- \( 1 \)
- \( -1 \)
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The Correct Option is D
Solution and Explanation
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:
\[
\boxed{-1}.
\]
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