# How do you prove  (sin+cos)(tan+cot)=sec+csc?

Oct 17, 2015

See explanation.

#### Explanation:

Prove: $\left(\sin x + \cos x\right) \left(\tan x + \cot x\right) = \sec x + \csc x$

The change I made in each step is colored red.

$\left[1\right] \textcolor{w h i t e}{X X} \left(\sin x + \cos x\right) \left(\tan x + \cot x\right)$

$\left[2\right] \textcolor{w h i t e}{X X} = \left(\sin x + \cos x\right) \left(\textcolor{red}{\sin \frac{x}{\cos} x} + \textcolor{red}{\cos \frac{x}{\sin} x}\right)$

$\left[3\right] \textcolor{w h i t e}{X X} = \textcolor{red}{{\sin}^{2} \frac{x}{\cos} x + \cos x + {\cos}^{2} \frac{x}{\sin} x + \sin x}$

$\left[4\right] \textcolor{w h i t e}{X X} = \frac{{\sin}^{2} x + \textcolor{red}{{\cos}^{2} x}}{\cos} x + \frac{{\cos}^{2} x + \textcolor{red}{{\sin}^{2} x}}{\sin} x$

$\left[5\right] \textcolor{w h i t e}{X X} = \frac{\textcolor{red}{1}}{\cos} x + \frac{\textcolor{red}{1}}{\sin} x$

$\left[6\right] \textcolor{w h i t e}{X X} = \textcolor{red}{\sec x} + \textcolor{red}{\csc x}$