Algebra Examples

$arccot (\sqrt{x}) + \arctan (x) = \frac{π}{2}$

Step 1

Solve for $\sqrt{x}$ .

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Step 1.1

Subtract $\arctan (x)$ from both sides of the equation.

$arccot (\sqrt{x}) = \frac{π}{2} - \arctan (x)$

Step 1.2

Rewrite the equation as $\frac{π}{2} - \arctan (x) = arccot (\sqrt{x})$ .

$\frac{π}{2} - \arctan (x) = arccot (\sqrt{x})$

Step 1.3

Subtract $\frac{π}{2}$ from both sides of the equation.

$- \arctan (x) = arccot (\sqrt{x}) - \frac{π}{2}$

Step 1.4

Divide each term in $- \arctan (x) = arccot (\sqrt{x}) - \frac{π}{2}$ by $- 1$ and simplify.

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Step 1.4.1

Divide each term in $- \arctan (x) = arccot (\sqrt{x}) - \frac{π}{2}$ by $- 1$ .

$\frac{- \arctan (x)}{- 1} = \frac{arccot (\sqrt{x})}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{\arctan (x)}{1} = \frac{arccot (\sqrt{x})}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.4.2.2

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

$\arctan (x) = \frac{arccot (\sqrt{x})}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Move the negative one from the denominator of $\frac{arccot (\sqrt{x})}{- 1}$ .

$\arctan (x) = - 1 \cdot arccot (\sqrt{x}) + \frac{- \frac{π}{2}}{- 1}$

Step 1.4.3.1.2

Rewrite $- 1 \cdot arccot (\sqrt{x})$ as $- arccot (\sqrt{x})$ .

$\arctan (x) = - arccot (\sqrt{x}) + \frac{- \frac{π}{2}}{- 1}$

Step 1.4.3.1.3

Dividing two negative values results in a positive value.

$\arctan (x) = - arccot (\sqrt{x}) + \frac{\frac{π}{2}}{1}$

Step 1.4.3.1.4

Divide $\frac{π}{2}$ by $1$ .

$\arctan (x) = - arccot (\sqrt{x}) + \frac{π}{2}$

Step 1.5

Take the inverse arctangent of both sides of the equation to extract $\sqrt{x}$ from inside the arctangent.

$x = \tan (- arccot (\sqrt{x}) + \frac{π}{2})$

Step 1.6

Rewrite the equation as $\tan (- arccot (\sqrt{x}) + \frac{π}{2}) = x$ .

$\tan (- arccot (\sqrt{x}) + \frac{π}{2}) = x$

Step 1.7

Take the inverse tangent of both sides of the equation to extract $arccot (\sqrt{x})$ from inside the tangent.

$- arccot (\sqrt{x}) + \frac{π}{2} = \arctan (x)$

Step 1.8

Subtract $\frac{π}{2}$ from both sides of the equation.

$- arccot (\sqrt{x}) = \arctan (x) - \frac{π}{2}$

Step 1.9

Divide each term in $- arccot (\sqrt{x}) = \arctan (x) - \frac{π}{2}$ by $- 1$ and simplify.

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Step 1.9.1

Divide each term in $- arccot (\sqrt{x}) = \arctan (x) - \frac{π}{2}$ by $- 1$ .

$\frac{- arccot (\sqrt{x})}{- 1} = \frac{\arctan (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.9.2

Simplify the left side.

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Step 1.9.2.1

Dividing two negative values results in a positive value.

$\frac{arccot (\sqrt{x})}{1} = \frac{\arctan (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.9.2.2

Divide $arccot (\sqrt{x})$ by $1$ .

$arccot (\sqrt{x}) = \frac{\arctan (x)}{- 1} + \frac{- \frac{π}{2}}{- 1}$

Step 1.9.3

Simplify the right side.

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Step 1.9.3.1

Simplify each term.

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Step 1.9.3.1.1

Move the negative one from the denominator of $\frac{\arctan (x)}{- 1}$ .

$arccot (\sqrt{x}) = - 1 \cdot \arctan (x) + \frac{- \frac{π}{2}}{- 1}$

Step 1.9.3.1.2

Rewrite $- 1 \cdot \arctan (x)$ as $- \arctan (x)$ .

$arccot (\sqrt{x}) = - \arctan (x) + \frac{- \frac{π}{2}}{- 1}$

Step 1.9.3.1.3

Dividing two negative values results in a positive value.

$arccot (\sqrt{x}) = - \arctan (x) + \frac{\frac{π}{2}}{1}$

Step 1.9.3.1.4

Divide $\frac{π}{2}$ by $1$ .

$arccot (\sqrt{x}) = - \arctan (x) + \frac{π}{2}$

Step 1.10

Take the inverse arccotangent of both sides of the equation to extract $\sqrt{x}$ from inside the arccotangent.

$\sqrt{x} = \cot (- \arctan (x) + \frac{π}{2})$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 3.2.1.2

Simplify.

$x = \cot^{2} (- \arctan (x) + \frac{π}{2})$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $\cot^{2} (- \arctan (x) + \frac{π}{2})$ from both sides of the equation.

$x - \cot^{2} (- \arctan (x) + \frac{π}{2}) = 0$

Step 4.2

Simplify each term.

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Step 4.2.1

Rearrange terms.

$x - \cot^{2} (\frac{π}{2} - \arctan (x)) = 0$

Step 4.2.2

Apply the difference of angles identity.

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

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

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

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

Step 4.2.3.2

Draw a triangle in the plane with vertices $(1, x)$ , $(1, 0)$ , and the origin. Then $\arctan (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, x)$ . Therefore, $\cot (\arctan (x))$ is $\frac{1}{x}$ .

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

Step 4.2.3.3

Multiply $0$ by $\frac{1}{x}$ .

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

Step 4.2.3.4

Add $0$ and $1$ .

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

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

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

Step 4.2.4.2

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

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

Step 4.2.4.3

Multiply $- 1$ by $0$ .

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

Step 4.2.4.4

Add $\frac{1}{x}$ and $0$ .

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

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.6

Multiply $x$ by $1$ .

$x - x^{2} = 0$

Step 4.3

Factor $x$ out of $x - x^{2}$ .

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Step 4.3.1

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

$x \cdot 1 - x^{2} = 0$

Step 4.3.2

Factor $x$ out of $- x^{2}$ .

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

Step 4.3.3

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$x (1 - x) = 0$

Step 4.4

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

$x = 0$

$1 - x = 0$

Step 4.5

Set $x$ equal to $0$ .

$x = 0$

Step 4.6

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 4.6.2

Solve $1 - x = 0$ for $x$ .

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Step 4.6.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4.6.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 4.6.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 4.6.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 4.6.2.2.3

Simplify the right side.

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Step 4.6.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 4.7

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

$x = 0, 1$

Step 5

Exclude the solutions that do not make $arccot (\sqrt{x}) + \arctan (x) = \frac{π}{2}$ true.

$x = 1$