I appreciate the help.
My attempt:
$$ \begin{align} \tan(x) + \cot(x) &= \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} \\ &= \frac{\sin^2(x)}{\cos(x) \sin(x)}+\frac{\cos^2(x)}{\cos(x) \sin(x)} \\ &= \frac{\sin^2(x)+\cos^2(x)}{\cos(x) \sin(x)}\\ &= \frac{1}{\cos(x) \sin(x)}\\ &= \frac{1}{\frac{1}{\sec(x)}\frac{1}{\csc(x)}}\\ &=\frac{1}{\frac{1}{\sec \csc}}\\ &=\frac{1}{1}\cdot \frac{\sec(x) \csc(x)}{1}\\ &= \sec(x) \csc(x) \end{align} $$