Which expression is equivalent to tan 2x - tan x ----------------- 1 + tan 2x tan x? (A) tan x (B) tan 3x (C) tan 2x - 1 --------------- 1 + tan 2x (D) \frac{tan x}{1 + tan 2x tan x} - HSC - SSCE Mathematics Extension 1 - Question 3 - 2016 - Paper 1

Recall the tangent of a sum formula: ( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b} ). This can help rearranging our expression.

Using the tangent addition formula, we can identify that: ( tan 2x = \tan(x + x) ), hence ( \tan 2x = \frac{2tan x}{1 - tan^2 x} ). This transforms our expression.

After simplification, we find that the expression equals ( \tan 3x ), where ( \tan 3x ) is the sum of ( \tan 2x ) and ( \tan x ).

Thus, the equivalent expression is (B) ( tan 3x ).