Precalculus Examples

$\sec (x) \cdot \sin (x) \cdot \cot (x) = 1$

Step 1

Start on the left side.

$\sec (x) \cdot \sin (x) \cdot \cot (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{1}{\cos (x)} \cdot \sin (x) \cdot \cot (x)$

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $\sin (x)$ out of $\frac{1}{\cos (x)} \cdot \sin (x)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

$\frac{1}{\cos (x)} \cdot \cos (x)$

Step 3.2

Combine $\frac{1}{\cos (x)}$ and $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)}$

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)}$

Step 3.3.2

Rewrite the expression.

$1$

Step 4

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

$\sec (x) \cdot \sin (x) \cdot \cot (x) = 1$ is an identity