Precalculus Examples

$\csc (x) \sec (x) \cot (x) = \csc^{2} (x)$

Step 1

Start on the left side.

$\csc (x) \sec (x) \cot (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{1}{\sin (x)} \sec (x) \cot (x)$

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 3.4

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{1} \sin (x)}$

Step 3.5

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{1} {\sin (x)}^{1}}$

Step 3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{\sin (x)}^{1 + 1}}$

Step 3.7

Add $1$ and $1$ .

$\frac{1}{\sin^{2} (x)}$

Step 4

Rewrite $\frac{1}{\sin^{2} (x)}$ as $\csc^{2} (x)$ .

$\csc^{2} (x)$

Step 5

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

$\csc (x) \sec (x) \cot (x) = \csc^{2} (x)$ is an identity