Precalculus Examples

$\frac{\cot (x)}{\sec (x)} = \csc (x) - \sin (x)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 2.2

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 2.4

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Divide $\cos (x)$ by $1$ .

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

Step 2.5.2

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

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Now consider the right side of the equation.

$\csc (x) - \sin (x)$

Step 5

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

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

Step 6

Write $- \sin (x)$ as a fraction with denominator $1$ .

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

Step 7

Add fractions.

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Step 7.1

To write $\frac{- \sin (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 7.2

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

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

Step 7.3

Combine the numerators over the common denominator.

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

Step 8

Multiply $- \sin (x) \sin (x)$ .

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

Step 9

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

$\frac{\cot (x)}{\sec (x)} = \csc (x) - \sin (x)$ is an identity