Precalculus Examples

$\frac{1}{\cot^{2} (x)} + \sec (x) \cos (x)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\cot^{2} (x)} + \sec (x) \cos (x)$

Step 1.2

Rewrite $\frac{1^{2}}{\cot^{2} (x)}$ as ${(\frac{1}{\cot (x)})}^{2}$ .

${(\frac{1}{\cot (x)})}^{2} + \sec (x) \cos (x)$

Step 1.3

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.5

Write $1$ as a fraction with denominator $1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Rewrite the expression.

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

Step 1.6.2

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

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

Step 1.7

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

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

Step 1.8

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.8.1

Rewrite $\sec (x) \cos (x)$ in terms of sines and cosines.

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

Step 1.8.2

Cancel the common factors.

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

Step 2

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 3

Apply pythagorean identity.

$\sec^{2} (x)$