Simplify $\frac{\cos 13°+\sin 13°}{\cos 13°-\sin 13°}$

The correct option is tan 58°.

We are given the trigonometric expression:

$\frac{\cos 13°+\sin 13°}{\cos 13°-\sin 13°}$

Step 1: Divide the numerator and denominator by $\cos 13°$.

Dividing every term in the expression by $\cos 13°$:

$=\frac{\frac{\cos 13°}{\cos 13°}+\frac{\sin 13°}{\cos 13°}}{\frac{\cos 13°}{\cos 13°}-\frac{\sin 13°}{\cos 13°}}$

Using the identity $\frac{\sin \theta }{\cos \theta }=\tan \theta$, we simplify this to:

$=\frac{1+\tan 13°}{1-\tan 13°}$

Step 2: Express 1 as a tangent value.

We know that $\tan 45°=1$. Substituting 1 with $\tan 45°$ in the expression:

$=\frac{\tan 45°+\tan 13°}{1-\tan 45°\cdot \tan 13°}$

Step 3: Apply the tangent addition formula.

Recall the compound angle formula for tangent:

$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\cdot \tan B}$

By comparing this formula with our expression where $A=45°$ and $B=13°$, we obtain:

$=\tan (45°+13°)$

$=\tan 58°$

Thus, the simplified expression is tan 58°.