Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{1}{1 + \sec (x)} + \frac{1}{1 - \sec (x)}$

Step 2

Add fractions.

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Step 2.1

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

$\frac{1}{1 + \sec (x)} \cdot \frac{1 - \sec (x)}{1 - \sec (x)} + \frac{1}{1 - \sec (x)}$

Step 2.2

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

$\frac{1}{1 + \sec (x)} \cdot \frac{1 - \sec (x)}{1 - \sec (x)} + \frac{1}{1 - \sec (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)}$

Step 2.3

Write each expression with a common denominator of $(1 + \sec (x)) (1 - \sec (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{1 + \sec (x)}$ by $\frac{1 - \sec (x)}{1 - \sec (x)}$ .

$\frac{1 - \sec (x)}{(1 + \sec (x)) (1 - \sec (x))} + \frac{1}{1 - \sec (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)}$

Step 2.3.2

Multiply $\frac{1}{1 - \sec (x)}$ by $\frac{1 + \sec (x)}{1 + \sec (x)}$ .

$\frac{1 - \sec (x)}{(1 + \sec (x)) (1 - \sec (x))} + \frac{1 + \sec (x)}{(1 - \sec (x)) (1 + \sec (x))}$

Step 2.3.3

Reorder the factors of $(1 - \sec (x)) (1 + \sec (x))$ .

$\frac{1 - \sec (x)}{(1 + \sec (x)) (1 - \sec (x))} + \frac{1 + \sec (x)}{(1 + \sec (x)) (1 - \sec (x))}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1 - \sec (x) + 1 + \sec (x)}{(1 + \sec (x)) (1 - \sec (x))}$

Step 3

Simplify numerator.

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Step 3.1

Add $1$ and $1$ .

$\frac{2 - \sec (x) + \sec (x)}{(1 + \sec (x)) (1 - \sec (x))}$

Step 3.2

Add $- \sec (x)$ and $\sec (x)$ .

$\frac{2 + 0}{(1 + \sec (x)) (1 - \sec (x))}$

Step 3.3

Add $2$ and $0$ .

$\frac{2}{(1 + \sec (x)) (1 - \sec (x))}$

Step 4

Simplify denominator.

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Step 4.1

Expand $(1 + \sec (x)) (1 - \sec (x))$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{2}{1 (1 - \sec (x)) + \sec (x) (1 - \sec (x))}$

Step 4.1.2

Apply the distributive property.

$\frac{2}{1 \cdot 1 + 1 (- \sec (x)) + \sec (x) (1 - \sec (x))}$

Step 4.1.3

Apply the distributive property.

$\frac{2}{1 \cdot 1 + 1 (- \sec (x)) + \sec (x) \cdot 1 + \sec (x) (- \sec (x))}$

Step 4.2

Simplify and combine like terms.

$\frac{2}{1 - \sec^{2} (x)}$

Step 5

Apply Pythagorean identity.

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Step 5.1

Reorder $1$ and $- \sec^{2} (x)$ .

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

Step 5.2

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

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

Step 5.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{2}{- \sec^{2} (x) - 1 \cdot - 1}$

Step 5.4

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

$\frac{2}{- (\sec^{2} (x) - 1)}$

Step 5.5

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

Multiply $2 (- \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}})$ .

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Step 7.2.1

Multiply $- 1$ by $2$ .

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

Step 7.2.2

Combine $- 2$ and $\frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ .

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

Step 7.3

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Combine.

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

Step 10

Multiply $2$ by $- 1$ .

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

Step 11

Multiply ${\sin (x)}^{2}$ by $1$ .

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

Step 12

Move the negative in front of the fraction.

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

Step 13

Now consider the right side of the equation.

$- 2 \cot^{2} (x)$

Step 14

Convert to sines and cosines.

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Step 14.1

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

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

Step 14.2

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

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

Step 15

Simplify.

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Step 15.1

Combine $- 2$ and $\frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ .

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

Step 15.2

Move the negative in front of the fraction.

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

Step 16

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

$\frac{1}{1 + \sec (x)} + \frac{1}{1 - \sec (x)} = - 2 \cot^{2} (x)$ is an identity