Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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

Step 4.2.2

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

$(\tan (x) + \sec (x)) (\tan (x) - \sec (x))$

Step 5

Expand $(\tan (x) + \sec (x)) (\tan (x) - \sec (x))$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\tan (x) (\tan (x) - \sec (x)) + \sec (x) (\tan (x) - \sec (x))$

Step 5.2

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sec (x)) + \sec (x) (\tan (x) - \sec (x))$

Step 5.3

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sec (x)) + \sec (x) \tan (x) + \sec (x) (- \sec (x))$

Step 6

Simplify terms.

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Step 6.1

Combine the opposite terms in $\tan (x) \tan (x) + \tan (x) (- \sec (x)) + \sec (x) \tan (x) + \sec (x) (- \sec (x))$ .

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Step 6.1.1

Reorder the factors in the terms $\tan (x) (- \sec (x))$ and $\sec (x) \tan (x)$ .

$\tan (x) \tan (x) - \sec (x) \tan (x) + \sec (x) \tan (x) + \sec (x) (- \sec (x))$

Step 6.1.2

Add $- \sec (x) \tan (x)$ and $\sec (x) \tan (x)$ .

$\tan (x) \tan (x) + 0 + \sec (x) (- \sec (x))$

Step 6.1.3

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

$\tan (x) \tan (x) + \sec (x) (- \sec (x))$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $\tan (x) \tan (x)$ .

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Add $1$ and $1$ .

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

Step 6.2.2

Rewrite using the commutative property of multiplication.

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.3.4

Add $1$ and $1$ .

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

Step 6.3

Simplify with factoring out.

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Step 6.3.1

Reorder $\tan^{2} (x)$ and $- \sec^{2} (x)$ .

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.3.4

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

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

Step 7

Apply pythagorean identity.

$- 1 \cdot 1$

Step 8

Multiply $- 1$ by $1$ .

$- 1$