Convert to Sine and Cosine

Express $\csc A$ and $\cot A$ in terms of $\sin A$ and $\cos A$.

$\csc A - \cot A = \frac{1}{\sin A} - \frac{\cos A}{\sin A} = \frac{1 - \cos A}{\sin A}$

Multiply the Factors

Substitute the rewritten term back into the original expression and combine.

$(1 + \cos A)\left(\frac{1 - \cos A}{\sin A}\right) = \frac{(1 + \cos A)(1 - \cos A)}{\sin A}$

Apply Algebraic Identity

Use the algebraic identity $(a + b)(a - b) = a^2 - b^2$ in the numerator.

$\frac{(1 + \cos A)(1 - \cos A)}{\sin A} = \frac{1 - \cos^2 A}{\sin A}$

Simplify using Trigonometric Identity

Substitute the Pythagorean identity $1 - \cos^2 A = \sin^2 A$ into the numerator and simplify.

$\frac{\sin^2 A}{\sin A} = \sin A$