Precalculus Examples

$\cot (θ) \cdot \sec (θ) = \csc (θ)$

Step 1

Start on the left side.

$\cot (θ) \cdot \sec (θ)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

Apply the reciprocal identity to $\sec (θ)$ .

$\frac{\cos (θ)}{\sin (θ)} \cdot \frac{1}{\cos (θ)}$

Step 3

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

$\frac{1}{\sin (θ)}$

Step 4

Rewrite $\frac{1}{\sin (θ)}$ as $\csc (θ)$ .

$\csc (θ)$

Step 5

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

$\cot (θ) \cdot \sec (θ) = \csc (θ)$ is an identity