Trigonometry Examples

$\frac{\cos (x)}{1 - \sin (x)} - \frac{1 + \sin (x)}{\cos (x)} = 0$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Add $1$ and $1$ .

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $- 1$ by $1$ .

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

Step 2.5.4

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

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Step 2.5.4.1

Apply the distributive property.

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

Step 2.5.4.2

Apply the distributive property.

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

Step 2.5.4.3

Apply the distributive property.

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

Step 2.5.5

Simplify and combine like terms.

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Step 2.5.5.1

Simplify each term.

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Step 2.5.5.1.1

Multiply $- 1$ by $1$ .

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

Step 2.5.5.1.2

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

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Step 2.5.5.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.5.1.2.2

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

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

Step 2.5.5.1.3

Multiply $- 1$ by $1$ .

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

Step 2.5.5.1.4

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

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Step 2.5.5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.5.1.4.2

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

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

Step 2.5.5.1.4.3

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

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

Step 2.5.5.1.4.4

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

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

Step 2.5.5.1.4.5

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

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

Step 2.5.5.1.4.6

Add $1$ and $1$ .

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

Step 2.5.5.2

Subtract $\sin (x)$ from $\sin (x)$ .

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

Step 2.5.5.3

Add $- 1$ and $0$ .

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

Step 2.5.6

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

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

Step 2.5.7

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

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

Step 2.5.8

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

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

Step 2.5.9

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

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

Step 2.5.10

Apply pythagorean identity.

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

Step 2.5.11

Subtract $\cos^{2} (x)$ from $\cos^{2} (x)$ .

$\frac{0}{\cos (x) (1 - \sin (x))}$

Step 2.6

Divide $0$ by $\cos (x) (1 - \sin (x))$ .

$0$

Step 3

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

$\frac{\cos (x)}{1 - \sin (x)} - \frac{1 + \sin (x)}{\cos (x)} = 0$ is an identity