Precalculus Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

Expand $(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ using the FOIL Method.

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Step 2.3.2.1

Apply the distributive property.

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

Step 2.3.2.2

Apply the distributive property.

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

Step 2.3.2.3

Apply the distributive property.

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

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

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

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Step 2.3.3.1.1.1

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

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

Step 2.3.3.1.1.2

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

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

Step 2.3.3.1.1.3

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

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

Step 2.3.3.1.1.4

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

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

Step 2.3.3.1.1.5

Add $1$ and $1$ .

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

Step 2.3.3.1.2

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

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Step 2.3.3.1.2.1

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

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

Step 2.3.3.1.2.2

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

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

Step 2.3.3.1.2.3

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

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

Step 2.3.3.1.2.4

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

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

Step 2.3.3.1.2.5

Add $1$ and $1$ .

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

Step 2.3.3.1.3

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

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Step 2.3.3.1.3.1

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

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

Step 2.3.3.1.3.2

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

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

Step 2.3.3.1.3.3

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

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

Step 2.3.3.1.3.4

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

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

Step 2.3.3.1.3.5

Add $1$ and $1$ .

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

Step 2.3.3.1.4

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

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Step 2.3.3.1.4.1

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

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

Step 2.3.3.1.4.2

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

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

Step 2.3.3.1.4.3

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

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

Step 2.3.3.1.4.4

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

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

Step 2.3.3.1.4.5

Add $1$ and $1$ .

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

Step 2.3.3.1.4.6

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

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

Step 2.3.3.1.4.7

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

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

Step 2.3.3.1.4.8

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

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

Step 2.3.3.1.4.9

Add $1$ and $1$ .

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

Step 2.3.3.2

Combine the numerators over the common denominator.

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

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

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

Step 2.3.6

Factor using the perfect square rule.

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

Apply the distributive property.

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

Step 2.4.2.2

Apply the distributive property.

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

Step 2.4.2.3

Apply the distributive property.

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

Step 2.4.3

Simplify and combine like terms.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sin (x) = 2 \cdot 1 \cdot \sin (x)$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = \sin (x)$ .

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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

Step 5.2

Cancel the common factor of ${(1 + \sin (x))}^{2}$ and $1 + \sin (x)$ .

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

Step 6

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

$\frac{1 + \sin (x)}{1 - \sin (x)} = {(\sec (x) + \tan (x))}^{2}$ is an identity