Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Write $\tan (b)$ as a fraction with denominator $1$ .

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

Step 3

Add fractions.

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Step 3.1

To write $\frac{\tan (b)}{1}$ as a fraction with a common denominator, multiply by $\frac{\tan (b)}{\tan (b)}$ .

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 4

Multiply $\tan (b) \tan (b)$ .

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

Step 5

Apply Pythagorean identity.

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Step 5.1

Rearrange terms.

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

Step 5.2

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

Apply the reciprocal identity to $\sec (b)$ .

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

Step 6.2

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

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

Step 6.3

Apply the product rule to $\frac{1}{\cos (b)}$ .

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

Combine.

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

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

Step 7.4

One to any power is one.

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

Step 8

Rewrite $\frac{1}{\cos (b) \sin (b)}$ as $\sec (b) \csc (b)$ .

$\sec (b) \csc (b)$

Step 9

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

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