4100.11 – Compound Fraction: Trig

We start at the bottom.

$$\begin{align*}\dfrac{\cos A}{\csc A} &= \dfrac{\quad \cos A \quad }{\dfrac{1}{\sin A}} \\[2em]&= \cos A \cdot \sin A \\ \\\dfrac{\cot A}{\cos A \cdot \sin A} &= \dfrac{\cos A}{\sin A} \cdot \dfrac{1}{\cos A \cdot \sin A} \\[1em]&= \dfrac{1}{{\sin^2}A} \\ \\\dfrac{\quad \sec A \quad }{\dfrac{1}{\sin^2 A}} &= \sec A \cdot{\sin^2 A} \\[1em]&= \dfrac{1}{\cos A} \cdot {\sin^2} A \\[1em]&= \dfrac{{\sin^2} A} {\cos A} \\ \\\dfrac{\quad \tan A \quad}{\dfrac{{\sin^2} A} {\cos A}} &= \dfrac{\sin A}{\cos A} \cdot \dfrac{\cos A}{{\sin^2} A} \\[2em]&= \dfrac{1}{\sin A} \\[1em] \\\dfrac{\quad \sin A \quad}{\dfrac{1}{\sin A}} &= {\sin^2 A}\end{align*}$$

After all that!