Simplify $$\frac{\cos 13^\circ + \sin 13^\circ}{\cos 13^\circ - \sin 13^\circ}$$​

Correct option is D

Given:
The expression to simplify is:

​$$\frac{\cos 13^\circ + \sin 13^\circ}{\cos 13^\circ - \sin 13^\circ}$$​

Formula Used:

​$$\frac{\cos A + \sin A}{\cos A - \sin A} = \tan\left( 45^\circ + A \right)$$  

Tan (A + B) = $$\frac{ tanA + tanB}{tanA - tanB}$$ 

tan ( 45+A)     = $$\frac{tan45^0 + tan A }{tan45^0 - tan A } = \frac{1+ tanA}{ 1 - tanA} = \frac{1 + \frac{sinA}{cosA} }{1 + \frac{sinA}{cosA}}= \frac{cosA + sinA}{cos A - sinA}$$   ​

Solution:

where A = $$13^\circ.$$ 

Substituting $$A = 13^\circ$$​ into the identity:

​$$\frac{\cos 13^\circ + \sin 13^\circ}{\cos 13^\circ - \sin 13^\circ} = \tan\left( 45^\circ + 13^\circ \right) = \tan 58^\circ$$​

The simplified expression is $${\tan 58^\circ}$$​

Thus, the correct option is (d) $$\tan 58^\circ$$​​