Trigonometry Examples

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

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 $a = \sec (x)$ and $b = \cos (x)$ .

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

Step 1.2

Simplify.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Reorder the factors in the terms $\frac{1}{\cos (x)} (- \cos (x))$ and $\cos (x) \frac{1}{\cos (x)}$ .

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

Step 5.1.2

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

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

Step 5.1.3

Add $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ and $0$ .

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

Step 5.2

Simplify each term.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

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

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

Step 5.2.1.5

Add $1$ and $1$ .

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

Step 5.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.3

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

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Step 5.2.3.1

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

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

Step 5.2.3.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

Add $1$ and $1$ .

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

Step 5.3

Simplify terms.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Combine.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Add $2$ and $2$ .

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

Rewrite $\frac{\cos^{4} (x)}{\sin^{2} (x)}$ as ${(\frac{\cos^{2} (x)}{\sin (x)})}^{2}$ .

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

Step 11

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{\sin (x)}$ and $b = \frac{\cos^{2} (x)}{\sin (x)}$ .

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

Step 12

Simplify terms.

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Step 12.1

Simplify each term.

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Step 12.1.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 12.1.2

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

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

Step 12.1.3

Separate fractions.

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

Step 12.1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 12.1.5

Divide $\cos (x)$ by $1$ .

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

Step 12.2

Combine the numerators over the common denominator.

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

Step 13

Apply pythagorean identity.

$(\csc (x) + \cos (x) \cot (x)) \frac{\sin^{2} (x)}{\sin (x)}$

Step 14

Simplify terms.

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Step 14.1

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

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Step 14.1.1

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

$(\csc (x) + \cos (x) \cot (x)) \frac{\sin (x) \sin (x)}{\sin (x)}$

Step 14.1.2

Cancel the common factors.

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Step 14.1.2.1

Multiply by $1$ .

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

Step 14.1.2.2

Cancel the common factor.

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

Step 14.1.2.3

Rewrite the expression.

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

Step 14.1.2.4

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

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

Step 14.2

Apply the distributive property.

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

Step 15

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 15.1

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

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

Step 15.2

Cancel the common factors.

$1 + \cos (x) \cot (x) \sin (x)$