Trigonometry Examples

$\sec (x) (\sec (x) - \cos (x))$

Step 1

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Rewrite using the commutative property of multiplication.

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

Step 5.2

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.3

Simplify each term.

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 6

Apply pythagorean identity.

$\tan^{2} (x)$