Evaluate (1-cos(x)^2)(1+cot(x)^2)=0

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

Step 1

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

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

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

Step 2

Set $1 - \cos^{2} (x)$ equal to $0$ and solve for $x$ .

Set $1 - \cos^{2} (x)$ equal to $0$ .

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

Solve $1 - \cos^{2} (x) = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

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

Divide each term in $- \cos^{2} (x) = - 1$ by $- 1$ and simplify.

Divide each term in $- \cos^{2} (x) = - 1$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$\cos (x) = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

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

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

$\cos (x) = 1$

$\cos (x) = - 1$

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

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

$x = \arccos (1)$

Simplify the right side.

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

$x = 0$

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.

$x = 2 π - 0$

Subtract $0$ from $2 π$ .

$x = 2 π$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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 $n$

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

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

$x = \arccos (- 1)$

Simplify the right side.

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

$x = π$

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.

$x = 2 π - π$

Subtract $π$ from $2 π$ .

$x = π$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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 $n$

List all of the solutions.

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

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 3

Set $1 + \cot^{2} (x)$ equal to $0$ and solve for $x$ .

Set $1 + \cot^{2} (x)$ equal to $0$ .

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

Solve $1 + \cot^{2} (x) = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\cot (x) = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\cot (x) = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$\cot (x) = i$

Next, use the negative value of the $\pm$ to find the second solution.

$\cot (x) = - i$

The complete solution is the result of both the positive and negative portions of the solution.

$\cot (x) = i, - i$

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

$\cot (x) = i$

$\cot (x) = - i$

Solve for $x$ in $\cot (x) = i$ .

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

$x = arccot (i)$

The inverse cotangent of $arccot (i)$ is undefined.

Undefined

Solve for $x$ in $\cot (x) = - i$ .

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

$x = arccot (- i)$

The inverse cotangent of $arccot (- i)$ is undefined.

Undefined

List all of the solutions.

No solution

Step 4

The final solution is all the values that make $(1 - \cos^{2} (x)) (1 + \cot^{2} (x)) = 0$ true.

$x = π n$ , for any integer $n$

Evaluate (1-cos(x)^2)(1+cot(x)^2)=0

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