Trigonometry Examples

$\cos (2 \arcsin (x))$

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 $b = x$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Simplify.

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

Step 2.5

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

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Apply the distributive property.

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

Step 2.6

Simplify and combine like terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Multiply $1$ by $1$ .

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

Step 2.6.1.2

Multiply $- x$ by $1$ .

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

Step 2.6.1.3

Multiply $x$ by $1$ .

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

Step 2.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.6.1.5

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

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Step 2.6.1.5.1

Move $x$ .

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

Step 2.6.1.5.2

Multiply $x$ by $x$ .

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

Step 2.6.2

Add $- x$ and $x$ .

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

Step 2.6.3

Add $1$ and $0$ .

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

Step 2.7

The functions sine and arcsine are inverses.

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

Step 3

Subtract $x^{2}$ from $- x^{2}$ .

$1 - 2 x^{2}$