Make One Fraction

A sum with a fraction in it becomes one fraction over that fraction's bottom. The top then often holds a Pythagorean identity. Prove $\sin\left(x\right)+\frac{{\cos\left(x\right)}^{2}}{\sin\left(x\right)} = \csc\left(x\right)$. Write as one fraction over sin x: $\frac{{\sin\left(x\right)}^{2}+{\cos\left(x\right)}^{2}}{\sin\left(x\right)}$. The top is 1, so the fraction is $\frac{1}{\sin\left(x\right)}$, which is csc x.

To put sin x over sin x, multiply it by sin x on top and below: that is why the first term lands on top as sin²x. Writing sin x on top unsquared changes the value.