Precalculus Examples

$\frac{1}{\cos^{2} (40)} - \frac{1}{\cot^{2} (40)}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{1}{\cos^{2} (40)} - \frac{1^{2}}{\cot^{2} (40)}$

Step 1.1.2

Rewrite $\frac{1^{2}}{\cot^{2} (40)}$ as ${(\frac{1}{\cot (40)})}^{2}$ .

$\frac{1}{\cos^{2} (40)} - {(\frac{1}{\cot (40)})}^{2}$

Step 1.1.3

Rewrite $\cot (40)$ in terms of sines and cosines.

$\frac{1}{\cos^{2} (40)} - {(\frac{1}{\frac{\cos (40)}{\sin (40)}})}^{2}$

Step 1.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (40)}{\sin (40)}$ .

$\frac{1}{\cos^{2} (40)} - {(1 \frac{\sin (40)}{\cos (40)})}^{2}$

Step 1.1.5

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

$\frac{1}{\cos^{2} (40)} - {(\frac{1}{1} \cdot \frac{\sin (40)}{\cos (40)})}^{2}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Rewrite the expression.

$\frac{1}{\cos^{2} (40)} - {(1 \frac{\sin (40)}{\cos (40)})}^{2}$

Step 1.1.6.2

Multiply $\frac{\sin (40)}{\cos (40)}$ by $1$ .

$\frac{1}{\cos^{2} (40)} - {(\frac{\sin (40)}{\cos (40)})}^{2}$

Step 1.1.7

Apply the product rule to $\frac{\sin (40)}{\cos (40)}$ .

$\frac{1}{\cos^{2} (40)} - \frac{\sin^{2} (40)}{\cos^{2} (40)}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{1 - \sin^{2} (40)}{\cos^{2} (40)}$

Step 2

Apply pythagorean identity.

$\frac{\cos^{2} (40)}{\cos^{2} (40)}$

Step 3

Cancel the common factor of $\cos^{2} (40)$ .

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Step 3.1

Cancel the common factor.

$\frac{\cos^{2} (40)}{\cos^{2} (40)}$

Step 3.2

Rewrite the expression.

$1$