Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Add $1$ and $1$ .

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

Step 2.5.2

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

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Step 2.5.2.1

Apply the distributive property.

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

Step 2.5.2.2

Apply the distributive property.

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

Step 2.5.2.3

Apply the distributive property.

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

Step 2.5.3

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) \cdot 1 - 1 \cos (x) - 1 \cdot 1$ .

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Step 2.5.3.1

Reorder the factors in the terms $\cos (x) \cdot 1$ and $- 1 \cos (x)$ .

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

Step 2.5.3.2

Subtract $1 \cos (x)$ from $1 \cos (x)$ .

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

Step 2.5.3.3

Add $\cos (x) \cos (x)$ and $0$ .

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

Step 2.5.4

Simplify each term.

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Step 2.5.4.1

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

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Step 2.5.4.1.1

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

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

Step 2.5.4.1.2

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

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

Step 2.5.4.1.3

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

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

Step 2.5.4.1.4

Add $1$ and $1$ .

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

Step 2.5.4.2

Multiply $- 1$ by $1$ .

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

Step 2.5.5

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 2.5.6

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

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

Step 2.5.7

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

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

Step 2.5.8

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

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

Step 2.5.9

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

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

Step 2.5.10

Apply pythagorean identity.

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

Step 2.5.11

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

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

Step 2.6

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

$0$

Step 3

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

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