Trigonometry Examples

$\frac{\sec^{2} (x)}{\cot (x)} - \tan^{3} (x) = \tan (x)$

Start on the left side.

$\frac{\sec^{2} (x)}{\cot (x)} - \tan^{3} (x)$

Apply Pythagorean identity in reverse.

$\frac{\tan^{2} (x) + 1}{\cot (x)} - \tan^{3} (x)$

Convert to sines and cosines.

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

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

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

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

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

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

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

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

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

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

Simplify.

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

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

Apply the distributive property.

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

Combine.

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

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

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

To write $\frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Write each expression with a common denominator of ${\cos (x)}^{3}$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

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Apply the distributive property.

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

Subtract ${\sin (x)}^{3}$ from ${\sin (x)}^{3}$ .

$\frac{\sin (x) {\cos (x)}^{2} + 0}{{\cos (x)}^{3}}$

Add $\sin (x) {\cos (x)}^{2}$ and $0$ .

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

Cancel the common factor of ${\cos (x)}^{2}$ and ${\cos (x)}^{3}$ .

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

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

$\tan (x)$

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

$\frac{\sec^{2} (x)}{\cot (x)} - \tan^{3} (x) = \tan (x)$ is an identity