# How do you verify tan(theta)/cot(theta)=tan^2(theta)?

Nov 15, 2015

Use the definition of $\cot \left(\theta\right)$ as the inverse of $\tan \left(\theta\right)$
and the fact that dividing by $\frac{1}{x}$ is the same as multiplying by $x$

#### Explanation:

$\frac{\tan \left(\theta\right)}{\cot \left(\theta\right)}$

$\textcolor{w h i t e}{\text{XXX}} = \tan \left(\theta\right) \div \cot \left(\theta\right)$

$\textcolor{w h i t e}{\text{XXX}} = \tan \left(\theta\right) \div \frac{1}{\tan \left(\theta\right)}$ [by definition of $\cot \left(\theta\right)$]

$\textcolor{w h i t e}{\text{XXX}} = \tan \left(\theta\right) \times \tan \left(\theta\right)$ [since $a \times \frac{1}{x} \iff a \times x$]

$\textcolor{w h i t e}{\text{XXX}} = {\tan}^{2} \left(\theta\right)$