Precalculus Examples

$\csc^{2} (u) - \cos (u) \sec (u) = \cot^{2} (u)$

Step 1

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

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

Step 2

Reorder the polynomial.

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $\cot^{2} (u) - \cos (u) \sec (u) + 1$ .

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Step 3.1.1

Move $1$ .

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

Step 3.1.2

Rearrange terms.

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

Step 3.1.3

Apply pythagorean identity.

$\csc^{2} (u) - \cos (u) \sec (u) = \cot^{2} (u)$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

Simplify each term.

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Step 3.1.4.1.1

Rewrite $\csc (u)$ in terms of sines and cosines.

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

Step 3.1.4.1.2

Apply the product rule to $\frac{1}{\sin (u)}$ .

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

Step 3.1.4.1.3

One to any power is one.

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

Step 3.1.4.1.4

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.1.4.1.4.1

Add parentheses.

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

Step 3.1.4.1.4.2

Reorder $\cos (u)$ and $\sec (u)$ .

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

Step 3.1.4.1.4.3

Rewrite $- \cos (u) \sec (u)$ in terms of sines and cosines.

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

Step 3.1.4.1.4.4

Cancel the common factors.

$\frac{1}{\sin^{2} (u)} - 1 \cdot 1 = \cot^{2} (u)$

Step 3.1.4.1.5

Multiply $- 1$ by $1$ .

$\frac{1}{\sin^{2} (u)} - 1 = \cot^{2} (u)$

Step 3.1.4.2

Simplify each term.

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Step 3.1.4.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\sin^{2} (u)} - 1 = \cot^{2} (u)$

Step 3.1.4.2.2

Rewrite $\frac{1^{2}}{\sin^{2} (u)}$ as ${(\frac{1}{\sin (u)})}^{2}$ .

${(\frac{1}{\sin (u)})}^{2} - 1 = \cot^{2} (u)$

Step 3.1.4.2.3

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

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

Step 3.1.5

Apply pythagorean identity.

$\cot^{2} (u) = \cot^{2} (u)$

Step 4

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \cot (u) | = | \cot (u) |$

Step 5

Solve for $u$ .

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Step 5.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\cot (u) = \cot (u)$

$\cot (u) = - \cot (u)$

$- \cot (u) = \cot (u)$

$- \cot (u) = - \cot (u)$

Step 5.2

After simplifying, there are only two unique equations to be solved.

$\cot (u) = \cot (u)$

$\cot (u) = - \cot (u)$

Step 5.3

Solve $\cot (u) = \cot (u)$ for $u$ .

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Step 5.3.1

For the two functions to be equal, the arguments of each must be equal.

$u = u$

Step 5.3.2

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

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Step 5.3.2.1

Subtract $u$ from both sides of the equation.

$u - u = 0$

Step 5.3.2.2

Subtract $u$ from $u$ .

$0 = 0$

Step 5.3.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 5.4

Solve $\cot (u) = - \cot (u)$ for $u$ .

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Step 5.4.1

Move all terms containing $\cot (u)$ to the left side of the equation.

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Step 5.4.1.1

Add $\cot (u)$ to both sides of the equation.

$\cot (u) + \cot (u) = 0$

Step 5.4.1.2

Add $\cot (u)$ and $\cot (u)$ .

$2 \cot (u) = 0$

Step 5.4.2

Divide each term in $2 \cot (u) = 0$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 \cot (u) = 0$ by $2$ .

$\frac{2 \cot (u)}{2} = \frac{0}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 \cot (u)}{2} = \frac{0}{2}$

Step 5.4.2.2.1.2

Divide $\cot (u)$ by $1$ .

$\cot (u) = \frac{0}{2}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0$ by $2$ .

$\cot (u) = 0$

Step 5.4.3

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

$u = arccot (0)$

Step 5.4.4

Simplify the right side.

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Step 5.4.4.1

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

$u = \frac{π}{2}$

Step 5.4.5

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.

$u = π + \frac{π}{2}$

Step 5.4.6

Simplify $π + \frac{π}{2}$ .

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Step 5.4.6.1

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

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

Step 5.4.6.2

Combine fractions.

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Step 5.4.6.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 5.4.6.2.2

Combine the numerators over the common denominator.

$u = \frac{π \cdot 2 + π}{2}$

Step 5.4.6.3

Simplify the numerator.

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Step 5.4.6.3.1

Move $2$ to the left of $π$ .

$u = \frac{2 \cdot π + π}{2}$

Step 5.4.6.3.2

Add $2 π$ and $π$ .

$u = \frac{3 π}{2}$

Step 5.4.7

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

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Step 5.4.7.1

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

$\frac{π}{| b |}$

Step 5.4.7.2

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

$\frac{π}{| 1 |}$

Step 5.4.7.3

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

$\frac{π}{1}$

Step 5.4.7.4

Divide $π$ by $1$ .

$π$

Step 5.4.8

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

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

Step 6

Consolidate the answers.

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