Trigonometry Examples

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

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

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

Step 3.3.2.2

Add $1$ and $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

To write $\frac{\cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}}$ .

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

Step 3.7

To write $\frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.8

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

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Step 3.8.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{{\cos (x)}^{3}}{{\cos (x)}^{3}}$ .

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

Step 3.8.2

Multiply $\frac{\sin (x) (2 {\cos (x)}^{2} + {\sin (x)}^{2})}{{\cos (x)}^{3}}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.8.3

Reorder the factors of $\sin (x) {\cos (x)}^{3}$ .

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 3.10

Simplify the numerator.

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

Step 4

Apply Pythagorean identity.

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Step 4.1

Rearrange terms.

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

Step 4.2

Apply pythagorean identity.

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

Step 5

One to any power is one.

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

Step 6

Rewrite $\frac{1}{\cos^{3} (x) \sin (x)}$ as $\cot (x) \sec^{4} (x)$ .

$\cot (x) \sec^{4} (x)$

Step 7

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

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$ is an identity