Secant, Cosecant, Cotangent

Secant, cosecant and cotangent are 1 divided by functions you know: $\sec\left(x\right) = \frac{1}{\cos\left(x\right)}$, $\csc\left(x\right) = \frac{1}{\sin\left(x\right)}$, $\cot\left(x\right) = \frac{\cos\left(x\right)}{\sin\left(x\right)}$. To simplify, write everything with sines and cosines. For example:

$$\tan\left(x\right)\csc\left(x\right) = \frac{\sin\left(x\right)}{\cos\left(x\right)}\frac{1}{\sin\left(x\right)} = \frac{1}{\cos\left(x\right)}$$

The sines cancel, and $\frac{1}{\cos\left(x\right)}$ is $\sec\left(x\right)$. To divide by a fraction, turn it upside down and multiply: $\frac{a}{\frac{b}{c}} = a\frac{c}{b}$.