Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

Cancel the common factor.

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

Step 3.3.1.4

Rewrite the expression.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

Cancel the common factor.

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

Step 3.3.6.3

Rewrite the expression.

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

Add $1$ and $1$ .

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

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

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

Step 3.4.3

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

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Step 3.4.3.1

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

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

Step 3.4.3.2

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

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

Step 3.4.3.3

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

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

Step 3.4.3.4

Add $1$ and $1$ .

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

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

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

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

Step 3.4.4.3

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

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

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

Step 3.4.5

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

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Step 3.4.5.1

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

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Step 3.4.5.1.1

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

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

Step 3.4.5.1.2

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

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

Step 3.4.5.2

Add $- 2$ and $1$ .

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

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

Step 3.4.6

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

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Step 3.4.6.1

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

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

Step 3.4.6.2

Subtract $1$ from $3$ .

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

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

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Step 3.5.1.1

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

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

Step 3.5.1.2

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

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

Step 3.5.1.3

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

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

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

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

Step 3.5.2.3

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

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

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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

Step 3.5.3.4

Add $1$ and $1$ .

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

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

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

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

Step 3.5.4.3

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

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

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

Step 3.5.5

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

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Step 3.5.5.1

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

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Step 3.5.5.1.1

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

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

Step 3.5.5.1.2

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

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

Step 3.5.5.2

Add $- 2$ and $1$ .

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

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

Step 3.5.6

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

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Step 3.5.6.1

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

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

Step 3.5.6.2

Subtract $1$ from $3$ .

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

Step 3.6

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

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$

Step 4

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

$\tan^{2} (θ)$

Step 5

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

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