Question

If \(x + z = 2y\) and \(y = \frac{\pi}{4}\), what is \( \tan x \cdot \tan y \cdot \tan z \)?

• A. 1
• B. 0
• C. \( \sqrt{2} \)
• D. 2

Hint:

For equations involving angles and their tangents, use the sum identity for tangent and apply known values such as \( \tan \frac{\pi}{4} = 1 \) to simplify the expression.

Answer 1

The Correct Option is A

We are given that \( x + z = 2y \) and \( y = \frac{\pi}{4} \). First, let's substitute the value of \( y \): \[ x + z = 2 \times \frac{\pi}{4} = \frac{\pi}{2} \] Thus, \( x + z = \frac{\pi}{2} \). Now, we know that: \[ \tan(x + z) = \tan \left(\frac{\pi}{2}\right) \] Since \( \tan \left(\frac{\pi}{2}\right) \) is undefined, but we can use the formula for the tangent of a sum: \[ \tan(x + z) = \frac{\tan x + \tan z}{1 - \tan x \cdot \tan z} \] For the equation to hold, we need: \[ 1 - \tan x \cdot \tan z = 0 \quad \Rightarrow \quad \tan x \cdot \tan z = 1 \] Now, we multiply both sides by \( \tan y = 1 \) (since \( \tan \frac{\pi}{4} = 1 \)): \[ \tan x \cdot \tan y \cdot \tan z = 1 \times 1 = 1 \] Thus, the correct answer is \( 1 \).