Step 1: Write the given expression

Start with the given trigonometric expression from the problem statement.

$\sin A + \frac{\cos A}{\tan(90° - A)}$

Step 2: Apply the complementary angle identity

Recall the standard trigonometric identity for complementary angles: $\tan(90° - A) = \cot A$. Substitute this into the denominator.

$\tan(90° - A) = \cot A$

Step 3: Simplify the fraction term

Express $\cot A$ in terms of sine and cosine as $\cot A = \frac{\cos A}{\sin A}$. Then simplify the fraction $\frac{\cos A}{\frac{\cos A}{\sin A}} = \sin A$.

$\frac{\cos A}{\cot A} = \frac{\cos A}{\frac{\cos A}{\sin A}} = \sin A$

Step 4: Combine the terms

Substitute the simplified fraction back into the original expression: $\sin A + \sin A = 2\sin A$.

$\sin A + \sin A = 2\sin A$