Discussion
If \(x + z = 2y\) and \(y = \frac{\pi}{4}\) ,then tan \(x\) tan \(y\) tan \(z\)=
Answer: A
\(\tan x \tan y \tan z = \) 1.
With \(y = \frac{\pi}{4}\), \(\tan y = 1\) and \(x + z = 2y = \frac{\pi}{2}\), so \(z = \frac{\pi}{2} - x\).
\[ \begin{aligned} \tan z &= \tan\left(\tfrac{\pi}{2} - x\right) = \cot x \\ \tan x \tan y \tan z &= \tan x \cdot 1 \cdot \cot x = 1 \end{aligned} \]
The product is the constant 1, so it cannot equal \(\tan(x - y)\) or \(\tan(z - y)\), which vary with \(x\); nor 0 or 1/2.
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