Question

If \( \sin(y + z - x) \), \( \sin(z + x - y) \), and \( \sin(x + y - z) \) are in A.P., then

• A. \( 2 \tan y = \tan x - \tan z \)
• B. \( \tan y = \tan x + \tan z \)
• C. \( 2 \tan y = \tan x + \tan z \)
• D. \( \tan y = \tan x - \tan z \)

Hint:

In A.P. problems with trigonometric functions, simplify using known identities and apply the formula for the middle term being the average of the first and third terms.

Answer 1

The Correct Option is C

Step 1: Use the formula for an arithmetic progression (A.P.).
If the terms \( \sin(y + z - x) \), \( \sin(z + x - y) \), and \( \sin(x + y - z) \) are in A.P., then the middle term is the average of the first and third terms: \[ \sin(z + x - y) = \frac{\sin(y + z - x) + \sin(x + y - z)}{2} \]
Step 2: Simplify the trigonometric expressions.
Using trigonometric identities, we simplify the expressions and find that: \[ 2 \tan y = \tan x + \tan z \]
Step 3: Conclusion.
Thus, the correct answer is \( 2 \tan y = \tan x + \tan z \).