Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Add fractions.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 4

Multiply $\tan (x) \tan (x)$ .

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

Step 5

Apply Pythagorean identity.

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Step 5.1

Rearrange terms.

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

Step 5.2

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

Combine.

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

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

Step 7.4

One to any power is one.

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

Step 8

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

$\sec (x) \csc (x)$

Step 9

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

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