Determine if True tan(theta)cot(theta)=1

$\tan (θ) \cot (θ) = 1$

Step 1

Start on the left side.

$\tan (θ) \cot (θ)$

Step 2

Convert to sines and cosines.

Write $\tan (θ)$ in sines and cosines using the quotient identity.

$\frac{\sin (θ)}{\cos (θ)} \cot (θ)$

Write $\cot (θ)$ in sines and cosines using the quotient identity.

$\frac{\sin (θ)}{\cos (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 3

Simplify.

Cancel the common factor of $\sin (θ)$ .

Cancel the common factor.

$\frac{\sin (θ)}{\cos (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$

Rewrite the expression.

$\frac{1}{\cos (θ)} \cos (θ)$

Cancel the common factor of $\cos (θ)$ .

Cancel the common factor.

$\frac{1}{\cos (θ)} \cos (θ)$

Rewrite the expression.

$1$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\tan (θ) \cot (θ) = 1$ is an identity