Trigonometry Examples

$\cos (2 \arctan (x))$

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Multiply $\frac{1}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply $\frac{1}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 2.1.3.2

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 2.1.3.3

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

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

Step 2.1.3.6

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

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Step 2.1.3.6.1

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

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

Step 2.1.3.6.2

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

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

Step 2.1.3.6.3

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

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

Step 2.1.3.6.4

Cancel the common factor of $2$ .

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Step 2.1.3.6.4.1

Cancel the common factor.

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

Step 2.1.3.6.4.2

Rewrite the expression.

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

Step 2.1.3.6.5

Simplify.

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

Step 2.1.4

Apply the product rule to $\frac{\sqrt{1 + x^{2}}}{1 + x^{2}}$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

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

Step 2.1.5.4.2

Rewrite the expression.

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

Step 2.1.5.5

Simplify.

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

Step 2.1.6

Cancel the common factor of $1 + x^{2}$ and ${(1 + x^{2})}^{2}$ .

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Step 2.1.6.1

Multiply by $1$ .

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

Step 2.1.6.2

Cancel the common factors.

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Step 2.1.6.2.1

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{2}$ .

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

Step 2.1.6.2.2

Cancel the common factor.

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

Step 2.1.6.2.3

Rewrite the expression.

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

Step 2.1.7

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

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

Step 2.1.8

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 2.1.9

Combine and simplify the denominator.

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Step 2.1.9.1

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 2.1.9.2

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 2.1.9.3

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

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

Step 2.1.9.4

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

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

Step 2.1.9.5

Add $1$ and $1$ .

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

Step 2.1.9.6

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

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Step 2.1.9.6.1

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

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

Step 2.1.9.6.2

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

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

Step 2.1.9.6.3

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

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

Step 2.1.9.6.4

Cancel the common factor of $2$ .

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Step 2.1.9.6.4.1

Cancel the common factor.

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

Step 2.1.9.6.4.2

Rewrite the expression.

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

Step 2.1.9.6.5

Simplify.

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

Step 2.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.10.1

Apply the product rule to $\frac{x \sqrt{1 + x^{2}}}{1 + x^{2}}$ .

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

Step 2.1.10.2

Apply the product rule to $x \sqrt{1 + x^{2}}$ .

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

Step 2.1.11

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

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Step 2.1.11.1

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

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

Step 2.1.11.2

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

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

Step 2.1.11.3

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

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

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.

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

Step 2.1.11.4.2

Rewrite the expression.

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

Step 2.1.11.5

Simplify.

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

Step 2.1.12

Cancel the common factor of $1 + x^{2}$ and ${(1 + x^{2})}^{2}$ .

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Step 2.1.12.1

Factor $1 + x^{2}$ out of $x^{2} (1 + x^{2})$ .

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

Step 2.1.12.2

Cancel the common factors.

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Step 2.1.12.2.1

Factor $1 + x^{2}$ out of ${(1 + x^{2})}^{2}$ .

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

Step 2.1.12.2.2

Cancel the common factor.

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

Step 2.1.12.2.3

Rewrite the expression.

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

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 $b = x$ .

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