Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \tan (x)$ .

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

Step 1.2

Simplify.

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Step 1.2.1

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

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

Step 1.2.2

Factor $\sin (x)$ out of $\sin (x) + \frac{\sin (x)}{\cos (x)}$ .

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Step 1.2.2.1

Multiply by $1$ .

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

Step 1.2.2.2

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.2.2.3

Factor $\sin (x)$ out of $\sin (x) \cdot 1 + \sin (x) \frac{1}{\cos (x)}$ .

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

Step 1.2.3

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

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

Step 1.2.4

Factor $\sin (x)$ out of $\sin (x) - \frac{\sin (x)}{\cos (x)}$ .

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Step 1.2.4.1

Multiply by $1$ .

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

Step 1.2.4.2

Factor $\sin (x)$ out of $- \frac{\sin (x)}{\cos (x)}$ .

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

Step 1.2.4.3

Factor $\sin (x)$ out of $\sin (x) \cdot 1 + \sin (x) (- \frac{1}{\cos (x)})$ .

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

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

Step 1.2.5

Combine exponents.

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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

Step 1.2.5.3

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

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

Step 1.2.5.4

Add $1$ and $1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3

Combine $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ and $\sin^{2} (x)$ .

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

Step 4

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 4.1

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

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

Step 4.2

Add $2$ and $2$ .

$\frac{\sin^{2} (x) (1 + \frac{1}{\cos (x)}) (1 - \frac{1}{\cos (x)})}{\frac{\sin^{4} (x)}{\cos^{2} (x)}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\sin^{2} (x) (1 + \frac{1}{\cos (x)}) (1 - \frac{1}{\cos (x)}) \frac{\cos^{2} (x)}{\sin^{4} (x)}$

Step 6

Cancel the common factor of $\sin^{2} (x)$ .

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Step 6.1

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

$\sin^{2} (x) ((1 + \frac{1}{\cos (x)}) (1 - \frac{1}{\cos (x)})) \frac{\cos^{2} (x)}{\sin^{4} (x)}$

Step 6.2

Factor $\sin^{2} (x)$ out of $\sin^{4} (x)$ .

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $1$ by $1$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.1.4

Rewrite using the commutative property of multiplication.

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

Step 8.1.5

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

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Step 8.1.5.1

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

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

Step 8.1.5.2

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

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

Step 8.1.5.3

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

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

Step 8.1.5.4

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

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

Step 8.1.5.5

Add $1$ and $1$ .

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

Step 8.2

Add $- \frac{1}{\cos (x)}$ and $\frac{1}{\cos (x)}$ .

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

Step 8.3

Add $1$ and $0$ .

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

Step 9

Simplify terms.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ by $1$ .

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

Step 9.3

Cancel the common factor of $\cos^{2} (x)$ .

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Step 9.3.1

Move the leading negative in $- \frac{1}{\cos^{2} (x)}$ into the numerator.

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

Step 9.3.2

Cancel the common factor.

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

Step 9.3.3

Rewrite the expression.

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

Step 9.4

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

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

Step 9.5

Combine the numerators over the common denominator.

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

Step 9.6

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

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

Step 9.7

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

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

Step 9.8

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

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

Step 9.9

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

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

Step 9.10

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

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

Step 10

Apply pythagorean identity.

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

Step 11

Cancel the common factor of $\sin^{2} (x)$ .

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Step 11.1

Cancel the common factor.

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

Step 11.2

Divide $- 1$ by $1$ .

$- 1$