Trigonometry Examples

$\frac{1}{\tan (b)} + \tan (b) = \frac{\sec^{2} (b)}{\tan (b)}$

Step 1

Start on the right side.

$\frac{\sec^{2} (b)}{\tan (b)}$

Step 2

Apply Pythagorean identity in reverse.

$\frac{\tan^{2} (b) + 1}{\tan (b)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

$\frac{{(\frac{\sin (b)}{\cos (b)})}^{2} + 1}{\tan (b)}$

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine.

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

Step 4.4

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

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

Step 4.5

Simplify each term.

$\frac{\sin (b)}{\cos (b)} + \frac{\cos (b)}{\sin (b)}$

Step 5

Add fractions.

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Step 5.1

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

$\frac{\sin (b)}{\cos (b)} \cdot \frac{\sin (b)}{\sin (b)} + \frac{\cos (b)}{\sin (b)}$

Step 5.2

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

$\frac{\sin (b)}{\cos (b)} \cdot \frac{\sin (b)}{\sin (b)} + \frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (b)}{\cos (b)}$

Step 5.3

Write each expression with a common denominator of $\cos (b) \sin (b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

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

$\frac{\sin (b) \sin (b)}{\cos (b) \sin (b)} + \frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (b)}{\cos (b)}$

Step 5.3.2

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

$\frac{\sin (b) \sin (b)}{\cos (b) \sin (b)} + \frac{\cos (b) \cos (b)}{\sin (b) \cos (b)}$

Step 5.3.3

Reorder the factors of $\sin (b) \cos (b)$ .

$\frac{\sin (b) \sin (b)}{\cos (b) \sin (b)} + \frac{\cos (b) \cos (b)}{\cos (b) \sin (b)}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{\sin (b) \sin (b) + \cos (b) \cos (b)}{\cos (b) \sin (b)}$

Step 6

Simplify each term.

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

Step 7

Reorder terms.

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

Step 8

Now consider the left side of the equation.

$\frac{1}{\tan (b)} + \tan (b)$

Step 9

Convert to sines and cosines.

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Step 9.1

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

$\frac{1}{\frac{\sin (b)}{\cos (b)}} + \tan (b)$

Step 9.2

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

$\frac{1}{\frac{\sin (b)}{\cos (b)}} + \frac{\sin (b)}{\cos (b)}$

Step 10

Simplify each term.

$\frac{\cos (b)}{\sin (b)} + \frac{\sin (b)}{\cos (b)}$

Step 11

Add fractions.

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Step 11.1

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

$\frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (b)}{\cos (b)} + \frac{\sin (b)}{\cos (b)}$

Step 11.2

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

$\frac{\cos (b)}{\sin (b)} \cdot \frac{\cos (b)}{\cos (b)} + \frac{\sin (b)}{\cos (b)} \cdot \frac{\sin (b)}{\sin (b)}$

Step 11.3

Write each expression with a common denominator of $\sin (b) \cos (b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.1

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

$\frac{\cos (b) \cos (b)}{\sin (b) \cos (b)} + \frac{\sin (b)}{\cos (b)} \cdot \frac{\sin (b)}{\sin (b)}$

Step 11.3.2

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

$\frac{\cos (b) \cos (b)}{\sin (b) \cos (b)} + \frac{\sin (b) \sin (b)}{\cos (b) \sin (b)}$

Step 11.3.3

Reorder the factors of $\sin (b) \cos (b)$ .

$\frac{\cos (b) \cos (b)}{\cos (b) \sin (b)} + \frac{\sin (b) \sin (b)}{\cos (b) \sin (b)}$

Step 11.4

Combine the numerators over the common denominator.

$\frac{\cos (b) \cos (b) + \sin (b) \sin (b)}{\cos (b) \sin (b)}$

Step 12

Simplify each term.

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

Step 13

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

$\frac{1}{\tan (b)} + \tan (b) = \frac{\sec^{2} (b)}{\tan (b)}$ is an identity