Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply Pythagorean identity.

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Step 6.1

Factor $- 1$ out of $- \sec^{2} (x)$ .

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

Step 6.2

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.3

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

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

Step 6.4

Apply pythagorean identity.

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

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.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

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

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

Step 8.2.5

Add $1$ and $1$ .

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

Step 8.3

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

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

Step 8.4

Write each expression with a common denominator of ${\cos (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

Add $1$ and $1$ .

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

Step 8.5

Combine the numerators over the common denominator.

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.7

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

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Step 8.7.1

Move the leading negative in $- \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ into the numerator.

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

Step 8.7.2

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

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

Step 8.7.3

Cancel the common factor.

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

Step 8.7.4

Rewrite the expression.

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

Step 8.8

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

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Step 8.8.1

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

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

Step 8.8.2

Cancel the common factor.

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

Step 8.8.3

Rewrite the expression.

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

Step 8.9

Move the negative in front of the fraction.

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

Step 8.10

Apply the distributive property.

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

Step 8.11

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

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Step 8.11.1

Multiply $- 1$ by $- 1$ .

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

Step 8.11.2

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

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

Step 8.12

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

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

Step 9

Now consider the left side of the equation.

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

Step 10

Convert to sines and cosines.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

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

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Step 11.3.1

Cancel the common factor.

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

Step 11.3.2

Rewrite the expression.

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

Step 11.4

Rewrite $- 1 \frac{\cos (x)}{\sin (x)}$ as $- \frac{\cos (x)}{\sin (x)}$ .

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

Step 12

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

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