Precalculus Examples

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

Step 1

Write the expression using exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 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 $b = \sqrt{1 - x^{2}}$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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 $b = x$ .

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

Step 3.3

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

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

Step 3.4

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 $b = x$ .

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