Answer
Given $\tan\theta = \frac{1}{\cot\theta}$, two equivalent forms of this identity are $\cot\theta = \frac{1}{\tan\theta}$ and $\tan\theta * \cot\theta = 1$.
Work Step by Step
To find the two equivalent forms of the identity, we need to solve for the other two variables in the identity.
Solving for $\cot\theta$,
Multiplying both sides by $\frac{\cot\theta}{\tan\theta}$,
$\tan\theta * \frac{\cot\theta}{\tan\theta} = \frac{1}{\cot\theta}*\frac{\cot\theta}{\tan\theta}$
Since $\frac{\tan\theta}{\tan\theta} = \frac{\cot\theta}{\cot\theta} = 1$
$\cot\theta = \frac{1}{\tan\theta}$
Solving for 1,
Multiply both sides by $\cot\theta$,
$\tan\theta * \cot\theta = \frac{1}{\cot\theta} * \cot\theta$
Since $\frac{\cot\theta}{\cot\theta} = 1$
$\tan\theta * \cot\theta = 1$.
Therefore given $\tan\theta = \frac{1}{\cot\theta}$, two equivalent forms of this identity are $\cot\theta = \frac{1}{\tan\theta}$ and $\tan\theta * \cot\theta = 1$.
