Trigonometry Examples

$\frac{\sin^{2} (x) - \tan (x)}{\cos^{2} (x) - \cot (x)} = \tan^{2} (x)$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 2.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $\cos (x) \sin (x)$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

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

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

Step 3.3.1.2

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 3.3.1.3

Cancel the common factor.

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

Step 3.3.1.4

Rewrite the expression.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

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

Tap for more steps...

Step 3.3.6.1

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

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

Step 3.3.6.2

Cancel the common factor.

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

Step 3.3.6.3

Rewrite the expression.

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

Add $1$ and $1$ .

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

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Factor ${\sin (x)}^{3}$ out of $\cos (x) \sin (x) {\sin (x)}^{2} - {\sin (x)}^{2}$ .

Tap for more steps...

Step 3.4.1.1

Factor ${\sin (x)}^{3}$ out of $\cos (x) \sin (x) {\sin (x)}^{2}$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) - {\sin (x)}^{2}}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.1.2

Factor ${\sin (x)}^{3}$ out of $- {\sin (x)}^{2}$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) + {\sin (x)}^{3} (- {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.1.3

Factor ${\sin (x)}^{3}$ out of ${\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) + {\sin (x)}^{3} (- {\sin (x)}^{- 1})$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1} - {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.2

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

Tap for more steps...

Step 3.4.2.1

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x)) - {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.2.2

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x)) + {\sin (x)}^{- 2} (- \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.2.3

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.3

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

Tap for more steps...

Step 3.4.3.1

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) ({\sin (x)}^{1} \sin (x)) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.3.2

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) ({\sin (x)}^{1} {\sin (x)}^{1}) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.3.3

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) {\sin (x)}^{1 + 1} - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.3.4

Add $1$ and $1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) {\sin (x)}^{2} - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.4

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

Tap for more steps...

Step 3.4.4.1

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x)) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.4.2

Factor $\sin (x)$ out of $- \sin (x)$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x)) + \sin (x) \cdot - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.4.3

Factor $\sin (x)$ out of $\sin (x) (\cos (x) \sin (x)) + \sin (x) \cdot - 1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x) - 1)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} \sin (x) (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.5

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

Tap for more steps...

Step 3.4.5.1

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

Tap for more steps...

Step 3.4.5.1.1

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} {\sin (x)}^{1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.5.1.2

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2 + 1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.5.2

Add $- 2$ and $1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

$\frac{{\sin (x)}^{3} {\sin (x)}^{- 1} (\cos (x) \sin (x) - 1)}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.6

Multiply ${\sin (x)}^{3}$ by ${\sin (x)}^{- 1}$ by adding the exponents.

Tap for more steps...

Step 3.4.6.1

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

$\frac{{\sin (x)}^{3 - 1} (\cos (x) \sin (x) - 1)}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Step 3.4.6.2

Subtract $1$ from $3$ .

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

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

Factor ${\cos (x)}^{3}$ out of $\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}$ .

Tap for more steps...

Step 3.5.1.1

Factor ${\cos (x)}^{3}$ out of $\cos (x) \sin (x) {\cos (x)}^{2}$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) - {\cos (x)}^{2}}$

Step 3.5.1.2

Factor ${\cos (x)}^{3}$ out of $- {\cos (x)}^{2}$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) + {\cos (x)}^{3} (- {\cos (x)}^{- 1})}$

Step 3.5.1.3

Factor ${\cos (x)}^{3}$ out of ${\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) + {\cos (x)}^{3} (- {\cos (x)}^{- 1})$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1} - {\cos (x)}^{- 1})}$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x)) - {\cos (x)}^{- 1})}$

Step 3.5.2.2

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x)) + {\cos (x)}^{- 2} (- \cos (x)))}$

Step 3.5.2.3

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x) - \cos (x)))}$

Step 3.5.3

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

Tap for more steps...

Step 3.5.3.1

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1} \cos (x) \sin (x) - \cos (x)))}$

Step 3.5.3.2

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1} {\cos (x)}^{1} \sin (x) - \cos (x)))}$

Step 3.5.3.3

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1 + 1} \sin (x) - \cos (x)))}$

Step 3.5.3.4

Add $1$ and $1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{2} \sin (x) - \cos (x)))}$

Step 3.5.4

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

Tap for more steps...

Step 3.5.4.1

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x)) - \cos (x)))}$

Step 3.5.4.2

Factor $\cos (x)$ out of $- \cos (x)$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x)) + \cos (x) \cdot - 1))}$

Step 3.5.4.3

Factor $\cos (x)$ out of $\cos (x) (\cos (x) \sin (x)) + \cos (x) \cdot - 1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x) - 1)))}$

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} \cos (x) (\cos (x) \sin (x) - 1))}$

Step 3.5.5

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

Tap for more steps...

Step 3.5.5.1

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

Tap for more steps...

Step 3.5.5.1.1

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} {\cos (x)}^{1} (\cos (x) \sin (x) - 1))}$

Step 3.5.5.1.2

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2 + 1} (\cos (x) \sin (x) - 1))}$

Step 3.5.5.2

Add $- 2$ and $1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 1} (\cos (x) \sin (x) - 1))}$

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} {\cos (x)}^{- 1} (\cos (x) \sin (x) - 1)}$

Step 3.5.6

Multiply ${\cos (x)}^{3}$ by ${\cos (x)}^{- 1}$ by adding the exponents.

Tap for more steps...

Step 3.5.6.1

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3 - 1} (\cos (x) \sin (x) - 1)}$

Step 3.5.6.2

Subtract $1$ from $3$ .

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

Step 3.6

Cancel the common factor of $\cos (x) \sin (x) - 1$ .

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

Step 4

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

$\tan^{2} (x)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sin^{2} (x) - \tan (x)}{\cos^{2} (x) - \cot (x)} = \tan^{2} (x)$ is an identity