Question

your turn! choose the option that best answers the question. use the formula for the tangent of the difference of two angles to complete the identity: \\( \tan ( 2 \pi - x ) = ? \\) a. \\( \tan x \\) b. \\( - \tan x \\) c. \\( \frac { 1 - \tan x } { 1 + \tan x } \\)

Explanation:

Step1: Recall the tangent difference formula

The formula for \(\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}\). Here \(A = 2\pi\) and \(B=x\).

Step2: Substitute the values

Since \(\tan(2\pi)=0\), we substitute into the formula: \(\tan(2\pi - x)=\frac{\tan(2\pi)-\tan x}{1+\tan(2\pi)\tan x}\).

Step3: Simplify the expression

Substituting \(\tan(2\pi) = 0\) gives \(\frac{0-\tan x}{1+0\times\tan x}=-\tan x\).

Answer:

B. \(-\tan x\)