Trigonometry Examples

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

Step 1

Apply the difference of angles identity.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

The functions sine and arcsine are inverses.

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

Step 2.1.2

The functions cosine and arccosine are inverses.

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

Step 2.1.3

Multiply $x$ by $x$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

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 $b = x$ .

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.1.9

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 $b = x$ .

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

Step 2.1.10

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

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Add $1$ and $1$ .

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

Step 2.1.11

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

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Step 2.1.11.1

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

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

Step 2.1.11.2

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

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

Step 2.1.11.3

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

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

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

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

Step 2.1.11.4.2

Rewrite the expression.

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

Step 2.1.11.5

Simplify.

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

Step 2.1.12

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 2.1.12.1

Apply the distributive property.

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

Step 2.1.12.2

Apply the distributive property.

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

Step 2.1.12.3

Apply the distributive property.

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

Step 2.1.13

Simplify and combine like terms.

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Step 2.1.13.1

Simplify each term.

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Step 2.1.13.1.1

Multiply $1$ by $1$ .

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

Step 2.1.13.1.2

Multiply $- x$ by $1$ .

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

Step 2.1.13.1.3

Multiply $x$ by $1$ .

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

Step 2.1.13.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.13.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.13.1.5.1

Move $x$ .

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

Step 2.1.13.1.5.2

Multiply $x$ by $x$ .

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

Step 2.1.13.2

Add $- x$ and $x$ .

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

Step 2.1.13.3

Add $1$ and $0$ .

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

Step 2.1.14

Apply the distributive property.

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

Step 2.1.15

Multiply $- 1$ by $1$ .

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

Step 2.1.16

Multiply $- - x^{2}$ .

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Step 2.1.16.1

Multiply $- 1$ by $- 1$ .

$x^{2} - 1 + 1 x^{2}$

Step 2.1.16.2

Multiply $x^{2}$ by $1$ .

$x^{2} - 1 + x^{2}$

Step 2.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 1$