Precalculus Examples

$\frac{\tan (x)}{1 - \cos (x)} = \csc (x) (1 + \sec (x))$

Step 1

Start on the left side.

$\frac{\tan (x)}{1 - \cos (x)}$

Step 2

Multiply $\frac{\tan (x)}{1 - \cos (x)}$ by $\frac{1 + \cos (x)}{1 + \cos (x)}$ .

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

Step 3

Combine.

$\frac{\tan (x) (1 + \cos (x))}{(1 - \cos (x)) (1 + \cos (x))}$

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\tan (x)$ by $1$ .

$\frac{\tan (x) + \tan (x) \cos (x)}{(1 - \cos (x)) (1 + \cos (x))}$

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 - \cos (x)) (1 + \cos (x))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{\tan (x) + \tan (x) \cos (x)}{1 (1 + \cos (x)) - \cos (x) (1 + \cos (x))}$

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

$\frac{\tan (x) + \tan (x) \cos (x)}{1 - \cos^{2} (x)}$

Step 6

Apply pythagorean identity.

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

Step 7

Convert to sines and cosines.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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Step 8.1.1.1

Cancel the common factor.

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

Step 8.1.1.2

Rewrite the expression.

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

Step 8.1.2

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

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Step 8.1.2.1

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

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

Step 8.1.2.2

Multiply by $1$ .

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

Step 8.1.2.3

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

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

Step 8.1.3

Write $1$ as a fraction with a common denominator.

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

Step 8.1.4

Combine the numerators over the common denominator.

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

Step 8.2

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

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

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.4

Combine.

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

Step 8.5

Cancel the common factor of $\sin (x)$ and ${\sin (x)}^{2}$ .

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Step 8.5.1

Factor $\sin (x)$ out of $\sin (x) (1 + \cos (x)) \cdot 1$ .

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

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $\sin (x)$ out of $\cos (x) {\sin (x)}^{2}$ .

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

Step 8.5.2.2

Cancel the common factor.

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

Step 8.5.2.3

Rewrite the expression.

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

Step 8.6

Multiply $1 + \cos (x)$ by $1$ .

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

Step 9

Now consider the right side of the equation.

$\csc (x) (1 + \sec (x))$

Step 10

Convert to sines and cosines.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Add fractions.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

Combine the numerators over the common denominator.

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

Step 13

Reorder terms.

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

Step 14

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

$\frac{\tan (x)}{1 - \cos (x)} = \csc (x) (1 + \sec (x))$ is an identity