Recall the identity for reciprocal

We know that the reciprocal of $\tan A$ is $\cot A$, so the expression becomes $\tan A + \cot A$.

$\frac{1}{\tan A} = \cot A$

Convert to Sine and Cosine

Substitute $\tan A = \frac{\sin A}{\cos A}$ and $\cot A = \frac{\cos A}{\sin A}$ into the expression.

$\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}$

Simplify using a common denominator

Combine the fractions using the denominator $(\sin A \cos A)$.

$\frac{\sin^2 A + \cos^2 A}{\sin A \cos A}$

Apply the Pythagorean identity

Since $\sin^2 A + \cos^2 A = 1$, the expression simplifies to $\frac{1}{\sin A \cos A}$.

$\frac{1}{\sin A \cos A} = \csc A \sec A$