Precalculus Examples

$\cos (\tan^{-1} (u) - \cos^{-1} (v))$

Step 1

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\cos (\tan^{-1} (u)) \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Draw a triangle in the plane with vertices $(1, u)$ , $(1, 0)$ , and the origin. Then $\tan^{-1} (u)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, u)$ . Therefore, $\cos (\tan^{-1} (u))$ is $\frac{1}{\sqrt{1 + u^{2}}}$ .

$\frac{1}{\sqrt{1 + u^{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.2

Multiply $\frac{1}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{1}{\sqrt{1 + u^{2}}} \cdot \frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply $\frac{1}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}} \sqrt{1 + u^{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.2

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{\sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.3

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{\sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.4

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

$\frac{\sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1 + 1}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{2}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6

Rewrite ${\sqrt{1 + u^{2}}}^{2}$ as $1 + u^{2}$ .

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Step 2.1.3.6.1

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

$\frac{\sqrt{1 + u^{2}}}{{({(1 + u^{2})}^{\frac{1}{2}})}^{2}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6.2

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

$\frac{\sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{1}{2} \cdot 2}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6.4

Cancel the common factor of $2$ .

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Step 2.1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{1 + u^{2}}}{{(1 + u^{2})}^{1}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.3.6.5

Simplify.

$\frac{\sqrt{1 + u^{2}}}{1 + u^{2}} \cos (\cos^{-1} (v)) + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.4

The functions cosine and arccosine are inverses.

$\frac{\sqrt{1 + u^{2}}}{1 + u^{2}} v + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.5

Combine $\frac{\sqrt{1 + u^{2}}}{1 + u^{2}}$ and $v$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \sin (\tan^{-1} (u)) \sin (\cos^{-1} (v))$

Step 2.1.6

Draw a triangle in the plane with vertices $(1, u)$ , $(1, 0)$ , and the origin. Then $\tan^{-1} (u)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, u)$ . Therefore, $\sin (\tan^{-1} (u))$ is $\frac{u}{\sqrt{1 + u^{2}}}$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u}{\sqrt{1 + u^{2}}} \sin (\cos^{-1} (v))$

Step 2.1.7

Multiply $\frac{u}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u}{\sqrt{1 + u^{2}}} \cdot \frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}} \sin (\cos^{-1} (v))$

Step 2.1.8

Combine and simplify the denominator.

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Step 2.1.8.1

Multiply $\frac{u}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}} \sqrt{1 + u^{2}}} \sin (\cos^{-1} (v))$

Step 2.1.8.2

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}} \sin (\cos^{-1} (v))$

Step 2.1.8.3

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}} \sin (\cos^{-1} (v))$

Step 2.1.8.4

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

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1 + 1}} \sin (\cos^{-1} (v))$

Step 2.1.8.5

Add $1$ and $1$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{2}} \sin (\cos^{-1} (v))$

Step 2.1.8.6

Rewrite ${\sqrt{1 + u^{2}}}^{2}$ as $1 + u^{2}$ .

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Step 2.1.8.6.1

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

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{({(1 + u^{2})}^{\frac{1}{2}})}^{2}} \sin (\cos^{-1} (v))$

Step 2.1.8.6.2

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

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{1}{2} \cdot 2}} \sin (\cos^{-1} (v))$

Step 2.1.8.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \sin (\cos^{-1} (v))$

Step 2.1.8.6.4

Cancel the common factor of $2$ .

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Step 2.1.8.6.4.1

Cancel the common factor.

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \sin (\cos^{-1} (v))$

Step 2.1.8.6.4.2

Rewrite the expression.

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{1}} \sin (\cos^{-1} (v))$

Step 2.1.8.6.5

Simplify.

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sin (\cos^{-1} (v))$

Step 2.1.9

Draw a triangle in the plane with vertices $(v, \sqrt{1^{2} - v^{2}})$ , $(v, 0)$ , and the origin. Then $\cos^{-1} (v)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(v, \sqrt{1^{2} - v^{2}})$ . Therefore, $\sin (\cos^{-1} (v))$ is $\sqrt{1 - v^{2}}$ .

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{1 - v^{2}}$

Step 2.1.10

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

$\frac{\sqrt{1 + u^{2}} v}{1 + u^{2}} + \frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{1^{2} - v^{2}}$

Step 2.1.11

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 $b = v$ .

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

Step 2.1.12

Multiply $\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{(1 + v) (1 - v)}$ .

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Step 2.1.12.1

Combine $\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}}$ and $\sqrt{(1 + v) (1 - v)}$ .

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

Step 2.1.12.2

Combine using the product rule for radicals.

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

Step 2.2

Combine the numerators over the common denominator.

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