Calculus Examples

$\cot (\arccos (x))$

Step 1

Draw a triangle in the plane with vertices

$(x, \sqrt{1^{2} - x^{2}})$ ,

$(x, 0)$ , and the origin. Then

$\arccos (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(x, \sqrt{1^{2} - x^{2}})$ . Therefore,

$\cot (\arccos (x))$ is

$\frac{x}{\sqrt{1 - x^{2}}}$ .

$\frac{x}{\sqrt{1 - x^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

$1$ as

$1^{2}$ .

$\frac{x}{\sqrt{1^{2} - x^{2}}}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = x$ .

$\frac{x}{\sqrt{(1 + x) (1 - x)}}$

Step 3

Multiply

$\frac{x}{\sqrt{(1 + x) (1 - x)}}$ by

$\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{x}{\sqrt{(1 + x) (1 - x)}} \cdot \frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

$\frac{x}{\sqrt{(1 + x) (1 - x)}}$ by

$\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{x \sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)} \sqrt{(1 + x) (1 - x)}}$

Step 4.2

Raise

$\sqrt{(1 + x) (1 - x)}$ to the power of

$1$ .

$\frac{x \sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} \sqrt{(1 + x) (1 - x)}}$

Step 4.3

Raise

$\sqrt{(1 + x) (1 - x)}$ to the power of

$1$ .

$\frac{x \sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} {\sqrt{(1 + x) (1 - x)}}^{1}}$

Step 4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x \sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1 + 1}}$

Step 4.5

Add

$1$ and

$1$ .

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

Step 4.6

Rewrite

${\sqrt{(1 + x) (1 - x)}}^{2}$ as

$(1 + x) (1 - x)$ .

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Step 4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{(1 + x) (1 - x)}$ as

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

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

Step 4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{x \sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 4.6.4

Cancel the common factor of

$2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

$\frac{x \sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{1}}$

Step 4.6.5

Simplify.

$\frac{x \sqrt{(1 + x) (1 - x)}}{(1 + x) (1 - x)}$