Algebra Examples

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

Step 1

Simplify with factoring out.

Tap for more steps...

Step 1.1

Move $- \cos^{2} (θ)$ .

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

Step 1.2

Factor $- 1$ out of $- \sin^{2} (θ)$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Apply pythagorean identity.

$\frac{- 1 \cdot 1 + \sec^{2} (θ)}{\tan (θ)}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Reorder $- 1^{2}$ and $\sec^{2} (θ)$ .

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

Step 3.3

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

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

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

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

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

Step 4

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

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

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

Tap for more steps...

Step 7.1.1.1

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

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

Step 7.1.1.2

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

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

Step 7.1.1.3

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

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

Step 7.1.1.4

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

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

Step 7.1.1.5

Add $1$ and $1$ .

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

Step 7.1.2

Combine $\frac{1}{\cos (θ)}$ and $- 1$ .

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

Step 7.1.3

Move the negative in front of the fraction.

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

Step 7.1.4

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

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

Step 7.1.5

Multiply $- 1$ by $1$ .

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

Step 7.2

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

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

Step 7.3

Add $\frac{1}{\cos^{2} (θ)}$ and $0$ .

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

Step 8

Simplify terms.

Tap for more steps...

Step 8.1

Apply the distributive property.

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

Step 8.2

Combine.

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

Step 8.3

Rewrite $- 1 \frac{\cos (θ)}{\sin (θ)}$ as $- \frac{\cos (θ)}{\sin (θ)}$ .

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

Step 8.4

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

Tap for more steps...

Step 8.4.1

Factor $\cos (θ)$ out of $1 \cos (θ)$ .

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

Step 8.4.2

Cancel the common factors.

Tap for more steps...

Step 8.4.2.1

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

$\frac{\cos (θ) \cdot 1}{\cos (θ) (\cos (θ) \sin (θ))} - \frac{\cos (θ)}{\sin (θ)}$

Step 8.4.2.2

Cancel the common factor.

$\frac{\cos (θ) \cdot 1}{\cos (θ) (\cos (θ) \sin (θ))} - \frac{\cos (θ)}{\sin (θ)}$

Step 8.4.2.3

Rewrite the expression.

$\frac{1}{\cos (θ) \sin (θ)} - \frac{\cos (θ)}{\sin (θ)}$

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

Separate fractions.

$\frac{1}{\sin (θ)} \cdot \frac{1}{\cos (θ)} - \frac{\cos (θ)}{\sin (θ)}$

Step 9.2

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

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

Step 9.3

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

$\csc (θ) \sec (θ) - \frac{\cos (θ)}{\sin (θ)}$

Step 9.4

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

$\csc (θ) \sec (θ) - \cot (θ)$