Calculus Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

Apply the product rule to $\frac{\sin (θ)}{\cos (θ)}$ .

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Combine.

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

Step 2.5

Rewrite using the commutative property of multiplication.

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

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Step 2.6.1.1

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

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

Step 2.6.1.2

Cancel the common factors.

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Step 2.6.1.2.1

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

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

Step 2.6.1.2.2

Cancel the common factor.

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

Step 2.6.1.2.3

Rewrite the expression.

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

Step 2.6.2

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

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Step 2.6.2.1

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

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

Step 2.6.2.2

Cancel the common factors.

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Step 2.6.2.2.1

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

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

Step 2.6.2.2.2

Cancel the common factor.

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

Step 2.6.2.2.3

Rewrite the expression.

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

Step 2.6.3

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

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Step 2.6.3.1

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

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

Step 2.6.3.2

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

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

Step 2.6.3.3

Cancel the common factor.

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

Step 2.6.3.4

Rewrite the expression.

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

Step 3

Write $- \cos (θ) \sin (θ)$ as a fraction with denominator $1$ .

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

Step 4

Add fractions.

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Step 4.1

To write $\frac{- \cos (θ) \sin (θ)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos (θ)}{\cos (θ)}$ .

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply Pythagorean identity in reverse.

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

Multiply by $1$ .

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

Step 7.1.1.2

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

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

Step 7.1.1.3

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

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Multiply $- 1$ by $1$ .

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

Step 7.1.4

Multiply $- - {\sin (θ)}^{2}$ .

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Step 7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.4.2

Multiply ${\sin (θ)}^{2}$ by $1$ .

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

Step 7.1.5

Subtract $1$ from $1$ .

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

Step 7.1.6

Add $0$ and ${\sin (θ)}^{2}$ .

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

Step 7.2

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

$\frac{\sin^{3} (θ)}{\cos (θ)}$

Step 8

Rewrite $\frac{\sin^{3} (θ)}{\cos (θ)}$ as $\tan (θ) \sin^{2} (θ)$ .

$\tan (θ) \sin^{2} (θ)$

Step 9

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

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