Trigonometry Examples

{cos}^{2} (x) - {sin}^{2} (x) = 0

$\cos^{2} (x) - \sin^{2} (x) = 0$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = cos (x)

$a = \cos (x)$ and

b = sin (x)

$b = \sin (x)$ .

(cos (x) + sin (x)) (cos (x) - sin (x)) = 0

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

cos (x) + sin (x) = 0

$\cos (x) + \sin (x) = 0$

cos (x) - sin (x) = 0

$\cos (x) - \sin (x) = 0$

Step 3

Set

cos (x) + sin (x)

$\cos (x) + \sin (x)$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.1

Set

cos (x) + sin (x)

$\cos (x) + \sin (x)$ equal to

0

$0$ .

cos (x) + sin (x) = 0

$\cos (x) + \sin (x) = 0$

Step 3.2

Solve

cos (x) + sin (x) = 0

$\cos (x) + \sin (x) = 0$ for

x

$x$ .

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Step 3.2.1

Divide each term in the equation by

cos (x)

$\cos (x)$ .

\frac{cos (x)}{cos (x)} + \frac{sin (x)}{cos (x)} = \frac{0}{cos (x)}

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.2.2

Cancel the common factor of

cos (x)

$\cos (x)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.2.2.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.2.3

Convert from

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

$\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 3.2.4

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 3.2.5

Convert from

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

$\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 3.2.6

Divide

$0$ by

$1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 3.2.7

Multiply

$0$ by

$\sec (x)$ .

$1 + \tan (x) = 0$

Step 3.2.8

Subtract

$1$ from both sides of the equation.

$\tan (x) = - 1$

Step 3.2.9

Take the inverse tangent of both sides of the equation to extract

$x$ from inside the tangent.

$x = \arctan (- 1)$

Step 3.2.10

Simplify the right side.

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Step 3.2.10.1

The exact value of

$\arctan (- 1)$ is

$- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 3.2.11

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from

$π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 3.2.12

Simplify the expression to find the second solution.

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Step 3.2.12.1

Add

$2 π$ to

$- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 3.2.12.2

The resulting angle of

$\frac{3 π}{4}$ is positive and coterminal with

$- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 3.2.13

Find the period of

$\tan (x)$ .

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Step 3.2.13.1

The period of the function can be calculated using

$\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 3.2.13.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 3.2.13.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{π}{1}$

Step 3.2.13.4

Divide

$π$ by

$1$ .

$π$

Step 3.2.14

Add

$π$ to every negative angle to get positive angles.

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Step 3.2.14.1

Add

$π$ to

$- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 3.2.14.2

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 3.2.14.3

Combine fractions.

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Step 3.2.14.3.1

Combine

$π$ and

$\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 3.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 3.2.14.4

Simplify the numerator.

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Step 3.2.14.4.1

Move

$4$ to the left of

$π$ .

$\frac{4 \cdot π - π}{4}$

Step 3.2.14.4.2

Subtract

$π$ from

$4 π$ .

$\frac{3 π}{4}$

Step 3.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 3.2.15

The period of the

$\tan (x)$ function is

$π$ so values will repeat every

$π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer

$n$

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer

$n$

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer

$n$

Step 4

Set

$\cos (x) - \sin (x)$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$\cos (x) - \sin (x)$ equal to

$0$ .

$\cos (x) - \sin (x) = 0$

Step 4.2

Solve

$\cos (x) - \sin (x) = 0$ for

$x$ .

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Step 4.2.1

Divide each term in the equation by

$\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2.2

Cancel the common factor of

$\cos (x)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2.2.2

Rewrite the expression.

$1 + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2.3

Separate fractions.

$1 + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2.4

Convert from

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

$\tan (x)$ .

$1 + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 4.2.5

Divide

$- 1$ by

$1$ .

$1 - \tan (x) = \frac{0}{\cos (x)}$

Step 4.2.6

Separate fractions.

$1 - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 4.2.7

Convert from

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

$\sec (x)$ .

$1 - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 4.2.8

Divide

$0$ by

$1$ .

$1 - \tan (x) = 0 \sec (x)$

Step 4.2.9

Multiply

$0$ by

$\sec (x)$ .

$1 - \tan (x) = 0$

Step 4.2.10

Subtract

$1$ from both sides of the equation.

$- \tan (x) = - 1$

Step 4.2.11

Divide each term in

$- \tan (x) = - 1$ by

$- 1$ and simplify.

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Step 4.2.11.1

Divide each term in

$- \tan (x) = - 1$ by

$- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 1}{- 1}$

Step 4.2.11.2

Simplify the left side.

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Step 4.2.11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 1}{- 1}$

Step 4.2.11.2.2

Divide

$\tan (x)$ by

$1$ .

$\tan (x) = \frac{- 1}{- 1}$

Step 4.2.11.3

Simplify the right side.

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Step 4.2.11.3.1

Divide

$- 1$ by

$- 1$ .

$\tan (x) = 1$

Step 4.2.12

Take the inverse tangent of both sides of the equation to extract

$x$ from inside the tangent.

$x = \arctan (1)$

Step 4.2.13

Simplify the right side.

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Step 4.2.13.1

The exact value of

$\arctan (1)$ is

$\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 4.2.14

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from

$π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 4.2.15

Simplify

$π + \frac{π}{4}$ .

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Step 4.2.15.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 4.2.15.2

Combine fractions.

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Step 4.2.15.2.1

Combine

$π$ and

$\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 4.2.15.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 4.2.15.3

Simplify the numerator.

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Step 4.2.15.3.1

Move

$4$ to the left of

$π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 4.2.15.3.2

Add

$4 π$ and

$π$ .

$x = \frac{5 π}{4}$

Step 4.2.16

Find the period of

$\tan (x)$ .

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Step 4.2.16.1

The period of the function can be calculated using

$\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 4.2.16.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 4.2.16.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{π}{1}$

Step 4.2.16.4

Divide

$π$ by

$1$ .

$π$

Step 4.2.17

The period of the

$\tan (x)$ function is

$π$ so values will repeat every

$π$ radians in both directions.

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer

$n$

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer

$n$

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer

$n$

Step 5

The final solution is all the values that make

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$ true.

$x = \frac{3 π}{4} + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer

$n$

Step 6

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer

$n$