Trigonometry Examples

$\cot (- x) \cos (- x) + \sin (- x) = - \csc (x)$

Step 1

Start on the left side.

$\cot (- x) \cos (- x) + \sin (- x)$

Step 2

Since $\cot (- x)$ is an odd function, rewrite $\cot (- x)$ as $- \cot (x)$ .

$- \cot (x) \cos (- x) + \sin (- x)$

Step 3

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$- \cot (x) \cos (x) + \sin (- x)$

Step 4

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$- \cot (x) \cos (x) - \sin (x)$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 5.2

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

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Step 5.2.1

Combine $\cos (x)$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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

Step 5.2.5

Add $1$ and $1$ .

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

Step 6

Apply Pythagorean identity in reverse.

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

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 $b = \sin (x)$ .

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

Step 7.2

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

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

Step 7.3

Combine $- \sin (x)$ and $\frac{\sin (x)}{\sin (x)}$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Simplify the numerator.

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Step 7.5.1

Factor $- 1$ out of $- (1 + \sin (x)) (1 - \sin (x)) - \sin (x) \sin (x)$ .

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Step 7.5.1.1

Factor $- 1$ out of $- (1 + \sin (x)) (1 - \sin (x))$ .

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

Step 7.5.1.2

Factor $- 1$ out of $- \sin (x) \sin (x)$ .

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

Step 7.5.1.3

Factor $- 1$ out of $- ((1 + \sin (x)) (1 - \sin (x))) - (\sin (x) \sin (x))$ .

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

Step 7.5.2

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

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Step 7.5.2.1

Apply the distributive property.

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

Step 7.5.2.2

Apply the distributive property.

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

Step 7.5.2.3

Apply the distributive property.

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

Step 7.5.3

Simplify and combine like terms.

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Step 7.5.3.1

Simplify each term.

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Step 7.5.3.1.1

Multiply $1$ by $1$ .

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

Step 7.5.3.1.2

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

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

Step 7.5.3.1.3

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

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

Step 7.5.3.1.4

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

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Step 7.5.3.1.4.1

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

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

Step 7.5.3.1.4.2

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

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

Step 7.5.3.1.4.3

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

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

Step 7.5.3.1.4.4

Add $1$ and $1$ .

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

Step 7.5.3.2

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

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

Step 7.5.3.3

Add $1$ and $0$ .

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

Step 7.5.4

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

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Step 7.5.4.1

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

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

Step 7.5.4.2

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

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

Step 7.5.4.3

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

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

Step 7.5.4.4

Add $1$ and $1$ .

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

Step 7.5.5

Add $- {\sin (x)}^{2}$ and ${\sin (x)}^{2}$ .

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

Step 7.5.6

Add $1$ and $0$ .

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

Step 7.6

Multiply $- 1$ by $1$ .

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

Step 7.7

Move the negative in front of the fraction.

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

Step 8

Write $- 1$ as a fraction with denominator $1$ .

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

Step 9

Combine.

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

Step 10

Multiply $- 1$ by $1$ .

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

Step 11

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

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

Step 12

Move the negative in front of the fraction.

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

Step 13

Now consider the right side of the equation.

$- \csc (x)$

Step 14

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

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

Step 15

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

$\cot (- x) \cos (- x) + \sin (- x) = - \csc (x)$ is an identity