The addition formula for the tangent function is given by:

To prove this addition formula, we begin with the well-known trigonometric identities for the sine and cosine of a sum:

Now, we can derive the tangent of the sum by dividing the sine identity by the cosine identity:

Next, we utilize the relationships $\tan x = \frac{\sin x}{\cos x}$ and $\tan y = \frac{\sin y}{\cos y}$. To simplify the expression, we can rewrite $\sin x$ and $\sin y$ in terms of $\tan x$ and $\tan y$:

Substituting these into our expression gives:

Factoring out $\cos x \cos y$ from both the numerator and denominator results in:

Thus, we have successfully derived the addition formula for the tangent function, confirming that:

This completes the proof of the addition formula for tangent.