What is (\tan(\pi+x)) equal to?

Correct answer: (\tan x)

The correct answer is option C: \\(\tan x\\). Tangent has period \\(\pi\\) radians, meaning adding \\(\pi\\) does not change its value: \\(\tan(x+\pi)=\tan x\\), wherever tangent is defined. This can also be seen from sine and cosine: both \\(\sin(x+\pi)=-\sin x\\) and \\(\cos(x+\pi)=-\cos x\\), so their ratio is \\(\frac{-\sin x}{-\cos x}=\frac{\sin x}{\cos x}=\tan x\\). Option A, \\(-\tan x\\), has the wrong sign. Option B, \\(\cot x\\), is the reciprocal of tangent, not the same function. Option D, \\(-\cot x\\), is also a different function. Option C matches the period identity. Exam cue: for tangent, adding or subtracting \\(\pi\\) leaves the value unchanged.