Trigonometry Examples

$\frac{1 + \sin (a)}{1 - \sin (a)} - \frac{1 - \sin (a)}{1 + \sin (a)} = 4 \tan (a) \sec (a)$

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $(1 - \sin (a)) (1 + \sin (a))$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Apply the distributive property.

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

Step 3.1.1.2

Apply the distributive property.

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

Step 3.1.1.3

Apply the distributive property.

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

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $1$ by $1$ .

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

Multiply $\sin (a) \sin (a)$ .

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Step 3.1.2.1.4.1

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

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

Step 3.1.2.1.4.2

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

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

Step 3.1.2.1.4.3

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

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

Step 3.1.2.1.4.4

Add $1$ and $1$ .

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

Step 3.1.2.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.1.5

Multiply $- - \sin (a)$ .

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.5.2

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

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

Step 3.1.6

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

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Step 3.1.6.1

Apply the distributive property.

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

Step 3.1.6.2

Apply the distributive property.

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

Step 3.1.6.3

Apply the distributive property.

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

Step 3.1.7

Simplify and combine like terms.

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Step 3.1.7.1

Simplify each term.

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Step 3.1.7.1.1

Multiply $- 1$ by $1$ .

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

Step 3.1.7.1.2

Multiply $- 1 (- \sin (a))$ .

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Step 3.1.7.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.7.1.2.2

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

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

Step 3.1.7.1.3

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

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

Step 3.1.7.1.4

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

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Step 3.1.7.1.4.1

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

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

Step 3.1.7.1.4.2

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

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

Step 3.1.7.1.4.3

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

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

Step 3.1.7.1.4.4

Add $1$ and $1$ .

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

Step 3.1.7.2

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

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

Step 3.2

Subtract $1$ from $1$ .

$\frac{2 \sin (a) + {\sin (a)}^{2} + 0 + 2 \sin (a) - {\sin (a)}^{2}}{(1 + \sin (a)) (1 - \sin (a))}$

Step 3.3

Add $2 \sin (a)$ and $0$ .

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

Step 3.4

Subtract ${\sin (a)}^{2}$ from ${\sin (a)}^{2}$ .

$\frac{2 \sin (a) + 2 \sin (a) + 0}{(1 + \sin (a)) (1 - \sin (a))}$

Step 3.5

Add $2 \sin (a)$ and $2 \sin (a)$ .

$\frac{4 \sin (a) + 0}{(1 + \sin (a)) (1 - \sin (a))}$

Step 3.6

Add $4 \sin (a)$ and $0$ .

$\frac{4 \sin (a)}{(1 + \sin (a)) (1 - \sin (a))}$

Step 4

Simplify denominator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

$\frac{4 \sin (a)}{1 (1 - \sin (a)) + \sin (a) (1 - \sin (a))}$

Step 4.1.2

Apply the distributive property.

$\frac{4 \sin (a)}{1 \cdot 1 + 1 (- \sin (a)) + \sin (a) (1 - \sin (a))}$

Step 4.1.3

Apply the distributive property.

$\frac{4 \sin (a)}{1 \cdot 1 + 1 (- \sin (a)) + \sin (a) \cdot 1 + \sin (a) (- \sin (a))}$

Step 4.2

Simplify and combine like terms.

$\frac{4 \sin (a)}{1 - \sin^{2} (a)}$

Step 5

Apply pythagorean identity.

$\frac{4 \sin (a)}{\cos^{2} (a)}$

Step 6

Rewrite $\frac{4 \sin (a)}{\cos^{2} (a)}$ as $4 \tan (a) \sec (a)$ .

$4 \tan (a) \sec (a)$

Step 7

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

$\frac{1 + \sin (a)}{1 - \sin (a)} - \frac{1 - \sin (a)}{1 + \sin (a)} = 4 \tan (a) \sec (a)$ is an identity