Question

If $( \sin(\alpha + \beta) = 1, \sin(\alpha - \beta) = \frac{1}{2}, \alpha, \beta \in [0, \pi/2] ), then ( \tan(\alpha + 2\beta) \cdot \tan(2\alpha + \beta) = ) $

• A. 1
• B. -1
• C. 0
• D. 4

Hint:

Check the quadrant carefully: $\tan(180-x) = -\tan x$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
First, determine the values of angles \( \alpha \) and \( \beta \) using the given trigonometric ratios.
Step 3: Detailed Explanation:
1. \( \sin(\alpha + \beta) = 1 \implies \alpha + \beta = 90^\circ \)
2. \( \sin(\alpha - \beta) = 1/2 \implies \alpha - \beta = 30^\circ \)
Adding the two equations: \( 2\alpha = 120^\circ \implies \alpha = 60^\circ \).
Subtracting: \( 2\beta = 60^\circ \implies \beta = 30^\circ \).
Now evaluate the expression:
\[ \alpha + 2\beta = 60^\circ + 60^\circ = 120^\circ \]
\[ 2\alpha + \beta = 120^\circ + 30^\circ = 150^\circ \]
\[ \tan(120^\circ) \cdot \tan(150^\circ) = (-\sqrt{3}) \cdot \left(-\frac{1}{\sqrt{3}}\right) = 1 \]
Step 4: Final Answer:
The product is 1.