Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

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

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

Step 6

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

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Add $1$ and $1$ .

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

Step 7.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.2.3

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

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

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

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

Step 7.2.3.4

Add $1$ and $1$ .

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

Step 8

Apply pythagorean identity.

$1$