Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Add fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $(1 + \cos (x)) (1 - \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(1 - \cos (x)) (1 + \cos (x))$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

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

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

Step 3.2

Reorder factors in $\tan (x) + \tan (x) (- \cos (x)) + \sin (x) + \sin (x) \cos (x)$ .

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

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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

Step 5

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

Move the leading negative in $- \frac{\sin (x)}{\cos (x)}$ into the numerator.

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

Step 7.1.1.2

Cancel the common factor.

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

Step 7.1.1.3

Rewrite the expression.

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

Step 7.1.2

Add $- \sin (x)$ and $\sin (x)$ .

$\frac{\frac{\sin (x)}{\cos (x)} + 0 + \sin (x) \cos (x)}{{\sin (x)}^{2}}$

Step 7.1.3

Add $\frac{\sin (x)}{\cos (x)}$ and $0$ .

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

Step 7.1.4

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

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Step 7.1.4.1

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

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

Step 7.1.4.2

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

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

Step 7.1.4.3

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

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

Step 7.1.5

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

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

Step 7.1.6

Combine the numerators over the common denominator.

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

Step 7.1.7

Multiply $\cos (x) \cos (x)$ .

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Step 7.1.7.1

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

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

Step 7.1.7.2

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

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

Step 7.1.7.3

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

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

Step 7.1.7.4

Add $1$ and $1$ .

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

Step 7.2

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

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

Step 7.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.4

Combine.

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

Step 7.5

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

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Step 7.5.1

Factor $\sin (x)$ out of $\sin (x) (1 + {\cos (x)}^{2}) \cdot 1$ .

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

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

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

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

Step 7.5.2.2

Cancel the common factor.

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

Step 7.5.2.3

Rewrite the expression.

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

Step 7.6

Multiply $1 + {\cos (x)}^{2}$ by $1$ .

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

Step 8

Now consider the right side of the equation.

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

Step 9

Convert to sines and cosines.

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Step 9.1

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

$\frac{\cos (x)}{\sin (x)} + \sec (x) \csc (x)$

Step 9.2

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

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

Step 9.3

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

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

Step 10

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

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

Step 11

Add fractions.

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Step 11.1

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

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

Step 11.2

Write each expression with a common denominator of $\sin (x) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.1

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

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

Step 11.2.2

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 11.3

Combine the numerators over the common denominator.

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

Step 12

Multiply $\cos (x) \cos (x)$ .

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

Step 13

Reorder terms.

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

Step 14

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

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