Precalculus Examples

$\cos (\arccos (x) + \arcsin (x))$

Step 1

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

$\cos (\arccos (x)) \cos (\arcsin (x)) - \sin (\arccos (x)) \sin (\arcsin (x))$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

The functions cosine and arccosine are inverses.

$x \cos (\arcsin (x)) - \sin (\arccos (x)) \sin (\arcsin (x))$

Step 2.1.2

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

$x \sqrt{1 - x^{2}} - \sin (\arccos (x)) \sin (\arcsin (x))$

Step 2.1.3

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

$x \sqrt{1^{2} - x^{2}} - \sin (\arccos (x)) \sin (\arcsin (x))$

Step 2.1.4

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 $b = x$ .

$x \sqrt{(1 + x) (1 - x)} - \sin (\arccos (x)) \sin (\arcsin (x))$

Step 2.1.5

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, $\sin (\arccos (x))$ is $\sqrt{1 - x^{2}}$ .

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

Step 2.1.6

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

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

Step 2.1.7

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 $b = x$ .

$x \sqrt{(1 + x) (1 - x)} - \sqrt{(1 + x) (1 - x)} \sin (\arcsin (x))$

Step 2.1.8

The functions sine and arcsine are inverses.

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

Step 2.2

Combine the opposite terms in $x \sqrt{(1 + x) (1 - x)} - \sqrt{(1 + x) (1 - x)} x$ .

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Step 2.2.1

Reorder the factors in the terms $x \sqrt{(1 + x) (1 - x)}$ and $- \sqrt{(1 + x) (1 - x)} x$ .

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

Step 2.2.2

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

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