Trigonometry Examples

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

Step 1

Subtract $\frac{\cos (x)}{1 + \sin (x)}$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $1$ by $1$ .

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

Step 2.5.2.1.2

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

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

Step 2.5.2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.5.2.1.4

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

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Step 2.5.2.1.4.1

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

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

Step 2.5.2.1.4.2

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

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

Step 2.5.2.1.4.3

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

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

Step 2.5.2.1.4.4

Add $1$ and $1$ .

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

Step 2.5.2.2

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

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

Step 2.5.2.3

Add $1$ and $0$ .

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

Step 2.5.3

Apply pythagorean identity.

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

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

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

Step 2.5.4.3

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

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

Step 2.5.4.4

Add $1$ and $1$ .

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

Step 2.5.5

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

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

Step 2.6

Separate fractions.

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

Step 2.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2.8

Divide $0$ by $1 + \sin (x)$ .

$0 \sec (x) = 0$

Step 2.9

Multiply $0$ by $\sec (x)$ .

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.