Trigonometry Examples

$\cos (2 \arccos (x))$

Step 1

Use the double-angle identity to transform $\cos (2 x)$ to $\cos^{2} (x) - \sin^{2} (x)$ .

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

Step 2

Simplify each term.

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Step 2.1

The functions cosine and arccosine are inverses.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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 $b = x$ .

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

Step 2.5

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

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Step 2.5.1

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 2.5.2

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 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 2.6

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

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.7

Simplify and combine like terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $1$ by $1$ .

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

Step 2.7.1.2

Multiply $- x$ by $1$ .

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

Step 2.7.1.3

Multiply $x$ by $1$ .

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

Step 2.7.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.7.1.5

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

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Step 2.7.1.5.1

Move $x$ .

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

Step 2.7.1.5.2

Multiply $x$ by $x$ .

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

Step 2.7.2

Add $- x$ and $x$ .

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

Step 2.7.3

Add $1$ and $0$ .

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

Step 2.8

Apply the distributive property.

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

Step 2.9

Multiply $- 1$ by $1$ .

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

Step 2.10

Multiply $- - x^{2}$ .

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.2

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

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

Step 3

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

$2 x^{2} - 1$