Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{\cos (x)}{\sec (x) - 1}$ .

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Step 1.1.1

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

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

Step 1.1.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{\cos (x) + 1}{\tan^{2} (x)}$ .

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Step 2.1.1

Simplify the denominator.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 2.1.4.2.1

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

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Step 2.1.4.2.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.4.2.2

Add $1$ and $2$ .

$\frac{\cos (x)}{\sec (x) - 1} = \frac{\cos^{3} (x)}{\sin^{2} (x)} + 1 \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Step 2.1.5

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

$\frac{\cos (x)}{\sec (x) - 1} = \frac{\cos^{3} (x)}{\sin^{2} (x)} + \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Step 2.1.6

Simplify each term.

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Step 2.1.6.1

Factor $\cos^{2} (x)$ out of $\cos^{3} (x)$ .

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

Step 2.1.6.2

Multiply by $1$ .

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

Step 2.1.6.3

Separate fractions.

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

Step 2.1.6.4

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

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

Step 2.1.6.5

Divide $\cos (x)$ by $1$ .

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

Step 2.1.6.6

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

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

Step 3

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 4

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

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

Step 5

Reorder the polynomial.

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

Step 6

Simplify $\csc^{2} (x) + \cot^{2} (x) - 1$ .

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Step 6.1

Move $- 1$ .

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

Step 6.2

Apply pythagorean identity.

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

Step 6.3

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

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

Step 7

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 8

Move all terms containing $x$ to the left side of the equation.

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Step 8.1

Subtract $2 \cot^{2} (x)$ from both sides of the equation.

$\frac{\cos (x)}{\sec (x) - 1} - 2 \cot^{2} (x) = 0$

Step 8.2

Simplify each term.

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Step 8.2.1

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

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} - 2 \cot^{2} (x) = 0$

Step 8.2.2

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

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} - 2 {(\frac{\cos (x)}{\sin (x)})}^{2} = 0$

Step 8.2.3

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

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} - 2 \frac{\cos^{2} (x)}{\sin^{2} (x)} = 0$

Step 8.2.4

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

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} + \frac{- 2 \cos^{2} (x)}{\sin^{2} (x)} = 0$

Step 8.2.5

Move the negative in front of the fraction.

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} - \frac{2 \cos^{2} (x)}{\sin^{2} (x)} = 0$

Step 8.3

To write $\frac{\cos (x)}{\frac{1}{\cos (x)} - 1}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)} - \frac{2 \cos^{2} (x)}{\sin^{2} (x)} = 0$

Step 8.4

To write $- \frac{2 \cos^{2} (x)}{\sin^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\frac{1}{\cos (x)} - 1}{\frac{1}{\cos (x)} - 1}$ .

$\frac{\cos (x)}{\frac{1}{\cos (x)} - 1} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)} - \frac{2 \cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\frac{1}{\cos (x)} - 1}{\frac{1}{\cos (x)} - 1} = 0$

Step 8.5

Write each expression with a common denominator of $(\frac{1}{\cos (x)} - 1) \sin^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.5.1

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

$\frac{\cos (x) \sin^{2} (x)}{(\frac{1}{\cos (x)} - 1) \sin^{2} (x)} - \frac{2 \cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{\frac{1}{\cos (x)} - 1}{\frac{1}{\cos (x)} - 1} = 0$

Step 8.5.2

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

$\frac{\cos (x) \sin^{2} (x)}{(\frac{1}{\cos (x)} - 1) \sin^{2} (x)} - \frac{2 \cos^{2} (x) (\frac{1}{\cos (x)} - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.5.3

Reorder the factors of $(\frac{1}{\cos (x)} - 1) \sin^{2} (x)$ .

$\frac{\cos (x) \sin^{2} (x)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} - \frac{2 \cos^{2} (x) (\frac{1}{\cos (x)} - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.6

Combine the numerators over the common denominator.

$\frac{\cos (x) \sin^{2} (x) - 2 \cos^{2} (x) (\frac{1}{\cos (x)} - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7

Simplify the numerator.

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Step 8.7.1

Factor $\cos (x)$ out of $\cos (x) \sin^{2} (x) - 2 \cos^{2} (x) (\frac{1}{\cos (x)} - 1)$ .

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Step 8.7.1.1

Factor $\cos (x)$ out of $\cos (x) \sin^{2} (x)$ .

$\frac{\cos (x) (\sin^{2} (x)) - 2 \cos^{2} (x) (\frac{1}{\cos (x)} - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.1.2

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

$\frac{\cos (x) (\sin^{2} (x)) + \cos (x) (- 2 \cos (x) (\frac{1}{\cos (x)} - 1))}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.1.3

Factor $\cos (x)$ out of $\cos (x) (\sin^{2} (x)) + \cos (x) (- 2 \cos (x) (\frac{1}{\cos (x)} - 1))$ .

$\frac{\cos (x) (\sin^{2} (x) - 2 \cos (x) (\frac{1}{\cos (x)} - 1))}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.2

Apply the distributive property.

$\frac{\cos (x) (\sin^{2} (x) - 2 \cos (x) \frac{1}{\cos (x)} - 2 \cos (x) \cdot - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.3

Cancel the common factor of $\cos (x)$ .

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Step 8.7.3.1

Factor $\cos (x)$ out of $- 2 \cos (x)$ .

$\frac{\cos (x) (\sin^{2} (x) + \cos (x) \cdot - 2 \frac{1}{\cos (x)} - 2 \cos (x) \cdot - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.3.2

Cancel the common factor.

$\frac{\cos (x) (\sin^{2} (x) + \cos (x) \cdot - 2 \frac{1}{\cos (x)} - 2 \cos (x) \cdot - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.3.3

Rewrite the expression.

$\frac{\cos (x) (\sin^{2} (x) - 2 - 2 \cos (x) \cdot - 1)}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.7.4

Multiply $- 1$ by $- 2$ .

$\frac{\cos (x) (\sin^{2} (x) - 2 + 2 \cos (x))}{\sin^{2} (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.8

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\frac{\cos (x) (\sin^{2} (x) - 2 + 2 \cos (x))}{\sin (x) \sin (x) (\frac{1}{\cos (x)} - 1)} = 0$

Step 8.9

Separate fractions.

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sin (x) (\frac{1}{\cos (x)} - 1)} \cdot \frac{\cos (x)}{\sin (x)} = 0$

Step 8.10

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sin (x) (\frac{1}{\cos (x)} - 1)} \cot (x) = 0$

Step 8.11

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sin (x) (\sec (x) - 1)} \cot (x) = 0$

Step 8.12

Combine $\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sin (x) (\sec (x) - 1)}$ and $\cot (x)$ .

$\frac{(\sin^{2} (x) - 2 + 2 \cos (x)) \cot (x)}{\sin (x) (\sec (x) - 1)} = 0$

Step 8.13

Separate fractions.

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} \cdot \frac{\cot (x)}{\sin (x)} = 0$

Step 8.14

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

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} \cdot \frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)} = 0$

Step 8.15

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\sin (x)}$ as a product.

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} (\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}) = 0$

Step 8.16

Simplify.

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Step 8.16.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} (\cot (x) \frac{1}{\sin (x)}) = 0$

Step 8.16.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} (\cot (x) \csc (x)) = 0$

Step 8.17

Multiply $\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1} (\cot (x) \csc (x))$ .

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Step 8.17.1

Combine $\cot (x)$ and $\frac{\sin^{2} (x) - 2 + 2 \cos (x)}{\sec (x) - 1}$ .

$\frac{\cot (x) (\sin^{2} (x) - 2 + 2 \cos (x))}{\sec (x) - 1} \csc (x) = 0$

Step 8.17.2

Combine $\frac{\cot (x) (\sin^{2} (x) - 2 + 2 \cos (x))}{\sec (x) - 1}$ and $\csc (x)$ .

$\frac{\cot (x) (\sin^{2} (x) - 2 + 2 \cos (x)) \csc (x)}{\sec (x) - 1} = 0$

Step 8.18

Reorder factors in $\frac{\cot (x) (\sin^{2} (x) - 2 + 2 \cos (x)) \csc (x)}{\sec (x) - 1}$ .

$\frac{\cot (x) \csc (x) (\sin^{2} (x) - 2 + 2 \cos (x))}{\sec (x) - 1} = 0$

Step 9

Set the numerator equal to zero.

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

Step 10

Solve the equation for $x$ .

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Step 10.1

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

$\cot (x) = 0$

$\csc (x) = 0$

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

Step 10.2

Set $\cot (x)$ equal to $0$ and solve for $x$ .

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Step 10.2.1

Set $\cot (x)$ equal to $0$ .

$\cot (x) = 0$

Step 10.2.2

Solve $\cot (x) = 0$ for $x$ .

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Step 10.2.2.1

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

$x = arccot (0)$

Step 10.2.2.2

Simplify the right side.

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Step 10.2.2.2.1

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

$x = \frac{π}{2}$

Step 10.2.2.3

The cotangent 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.

$x = π + \frac{π}{2}$

Step 10.2.2.4

Simplify $π + \frac{π}{2}$ .

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Step 10.2.2.4.1

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

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

Step 10.2.2.4.2

Combine fractions.

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Step 10.2.2.4.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 10.2.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 10.2.2.4.3

Simplify the numerator.

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Step 10.2.2.4.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 10.2.2.4.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 10.2.2.5

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

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Step 10.2.2.5.1

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

$\frac{π}{| b |}$

Step 10.2.2.5.2

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

$\frac{π}{| 1 |}$

Step 10.2.2.5.3

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

$\frac{π}{1}$

Step 10.2.2.5.4

Divide $π$ by $1$ .

$π$

Step 10.2.2.6

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

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

Step 10.3

Set $\csc (x)$ equal to $0$ and solve for $x$ .

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Step 10.3.1

Set $\csc (x)$ equal to $0$ .

$\csc (x) = 0$

Step 10.3.2

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 10.4

Set $\sin^{2} (x) - 2 + 2 \cos (x)$ equal to $0$ and solve for $x$ .

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Step 10.4.1

Set $\sin^{2} (x) - 2 + 2 \cos (x)$ equal to $0$ .

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

Step 10.4.2

Solve $\sin^{2} (x) - 2 + 2 \cos (x) = 0$ for $x$ .

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Step 10.4.2.1

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

$(1 - \cos^{2} (x)) - 2 + 2 \cos (x) = 0$

Step 10.4.2.2

Solve for $x$ .

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Step 10.4.2.2.1

Subtract $2$ from $1$ .

$- 1 - \cos^{2} (x) + 2 \cos (x) = 0$

Step 10.4.2.2.2

Factor by grouping.

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Step 10.4.2.2.2.1

Reorder terms.

$- \cos^{2} (x) + 2 \cos (x) - 1 = 0$

Step 10.4.2.2.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 10.4.2.2.2.2.1

Factor $2$ out of $2 \cos (x)$ .

$- \cos^{2} (x) + 2 \cos (x) - 1 = 0$

Step 10.4.2.2.2.2.2

Rewrite $2$ as $1$ plus $1$

$- \cos^{2} (x) + (1 + 1) \cos (x) - 1 = 0$

Step 10.4.2.2.2.2.3

Apply the distributive property.

$- \cos^{2} (x) + 1 \cos (x) + 1 \cos (x) - 1 = 0$

Step 10.4.2.2.2.2.4

Multiply $\cos (x)$ by $1$ .

$- \cos^{2} (x) + \cos (x) + 1 \cos (x) - 1 = 0$

Step 10.4.2.2.2.2.5

Multiply $\cos (x)$ by $1$ .

$- \cos^{2} (x) + \cos (x) + \cos (x) - 1 = 0$

Step 10.4.2.2.2.3

Factor out the greatest common factor from each group.

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Step 10.4.2.2.2.3.1

Group the first two terms and the last two terms.

$- \cos^{2} (x) + \cos (x) + \cos (x) - 1 = 0$

Step 10.4.2.2.2.3.2

Factor out the greatest common factor (GCF) from each group.

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

Step 10.4.2.2.2.4

Factor the polynomial by factoring out the greatest common factor, $- \cos (x) + 1$ .

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

Step 10.4.2.2.3

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

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

$\cos (x) - 1 = 0$

Step 10.4.2.2.4

Set $- \cos (x) + 1$ equal to $0$ and solve for $x$ .

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Step 10.4.2.2.4.1

Set $- \cos (x) + 1$ equal to $0$ .

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

Step 10.4.2.2.4.2

Solve $- \cos (x) + 1 = 0$ for $x$ .

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Step 10.4.2.2.4.2.1

Subtract $1$ from both sides of the equation.

$- \cos (x) = - 1$

Step 10.4.2.2.4.2.2

Divide each term in $- \cos (x) = - 1$ by $- 1$ and simplify.

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Step 10.4.2.2.4.2.2.1

Divide each term in $- \cos (x) = - 1$ by $- 1$ .

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

Step 10.4.2.2.4.2.2.2

Simplify the left side.

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Step 10.4.2.2.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.4.2.2.4.2.2.2.2

Divide $\cos (x)$ by $1$ .

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

Step 10.4.2.2.4.2.2.3

Simplify the right side.

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Step 10.4.2.2.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$\cos (x) = 1$

Step 10.4.2.2.4.2.3

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

$x = \arccos (1)$

Step 10.4.2.2.4.2.4

Simplify the right side.

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Step 10.4.2.2.4.2.4.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 10.4.2.2.4.2.5

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.

$x = 2 π - 0$

Step 10.4.2.2.4.2.6

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 10.4.2.2.4.2.7

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

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Step 10.4.2.2.4.2.7.1

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

$\frac{2 π}{| b |}$

Step 10.4.2.2.4.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 10.4.2.2.4.2.7.3

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

$\frac{2 π}{1}$

Step 10.4.2.2.4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.4.2.2.4.2.8

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

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 10.4.2.2.5

The final solution is all the values that make $(- \cos (x) + 1) (\cos (x) - 1) = 0$ true.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 10.5

The final solution is all the values that make $\cot (x) \csc (x) (\sin^{2} (x) - 2 + 2 \cos (x)) = 0$ true.

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

Step 11

Consolidate the answers.

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Step 11.1

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

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

Step 11.2

Consolidate $2 π n$ and $2 π + 2 π n$ to $2 π n$ .

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

Step 12

Exclude the solutions that do not make $\frac{\cos (x)}{\sec (x) - 1} = \frac{\cos (x) + 1}{\tan^{2} (x)}$ true.

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