Trigonometry Examples

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

Step 1

Replace the $- \sin^{2} (x)$ with $- (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$1 - 1 \cdot 1 + \cos^{2} (x) = - \cos (x)$

Step 2.2

Multiply $- 1$ by $1$ .

$1 - 1 + \cos^{2} (x) = - \cos (x)$

Step 2.3

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

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$1 - 1 + 1 \cos^{2} (x) = - \cos (x)$

Step 2.3.2

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

$1 - 1 + \cos^{2} (x) = - \cos (x)$

Step 3

Simplify by subtracting numbers.

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Step 3.1

Subtract $1$ from $1$ .

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

Step 3.2

Add $0$ and $\cos^{2} (x)$ .

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

Step 4

Substitute $u$ for $\cos (x)$ .

${(u)}^{2} = - (u)$

Step 5

Add $u$ to both sides of the equation.

$u^{2} + u = 0$

Step 6

Factor $u$ out of $u^{2} + u$ .

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Step 6.1

Factor $u$ out of $u^{2}$ .

$u \cdot u + u = 0$

Step 6.2

Raise $u$ to the power of $1$ .

$u \cdot u + u = 0$

Step 6.3

Factor $u$ out of $u^{1}$ .

$u \cdot u + u \cdot 1 = 0$

Step 6.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

$u (u + 1) = 0$

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u + 1 = 0$

Step 8

Set $u$ equal to $0$ .

$u = 0$

Step 9

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 10

The final solution is all the values that make $u (u + 1) = 0$ true.

$u = 0, - 1$

Step 11

Substitute $\cos (x)$ for $u$ .

$\cos (x) = 0, - 1$

Step 12

Set up each of the solutions to solve for $x$ .

$\cos (x) = 0$

$\cos (x) = - 1$

Step 13

Solve for $x$ in $\cos (x) = 0$ .

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Step 13.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (0)$

Step 13.2

Simplify the right side.

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Step 13.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 13.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 13.4

Simplify $2 π - \frac{π}{2}$ .

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Step 13.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 13.4.2

Combine fractions.

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Step 13.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 13.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 13.4.3

Simplify the numerator.

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Step 13.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 13.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 13.5

Find the period of $\cos (x)$ .

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Step 13.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 13.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 13.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 14

Solve for $x$ in $\cos (x) = - 1$ .

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Step 14.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- 1)$

Step 14.2

Simplify the right side.

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Step 14.2.1

The exact value of $\arccos (- 1)$ is $π$ .

$x = π$

Step 14.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - π$

Step 14.4

Subtract $π$ from $2 π$ .

$x = π$

Step 14.5

Find the period of $\cos (x)$ .

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Step 14.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 14.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 14.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = π + 2 π n$ , for any integer $n$

Step 15

List all of the solutions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, π + 2 π n$ , for any integer $n$

Step 16

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

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