Question

If \( x + z = 2y \) and \( y = \frac{\pi}{4} \), then \( \tan x \tan y \tan z = \)

• A. \(1 \)
• B. \( \tan(x-y) \)
• C. \( \tan(z-y) \)
• D. \( \frac{1}{2} \)
• E. \(0 \)

Hint:

If \( x+z = \frac{\pi}{2} \), then \( \tan x \tan z = 1 \).

Answer 1

The Correct Option is A

Concept: Use identity: \[ x + z = 2y \Rightarrow x + z = \frac{\pi}{2} \]

Step 1: Substitute value of \(y\).
\[ y = \frac{\pi}{4} \Rightarrow x + z = \frac{\pi}{2} \]

Step 2: Use identity.
\[ \tan x \tan z = 1 \quad \text{if } x+z=\frac{\pi}{2} \]

Step 3: Evaluate expression.
\[ \tan y = \tan \frac{\pi}{4} = 1 \] \[ \Rightarrow \tan x \tan y \tan z = 1 \cdot 1 = 1 \]