Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

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

Step 1.2

Remove the plus-minus sign on $\pm \sqrt{\frac{1 + \cos (x)}{2}}$ because it is raised to an even power.

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

Step 1.3

Rewrite $\sqrt{\frac{1 + \cos (x)}{2}}$ as $\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ .

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

Step 1.4

Multiply $\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{\sqrt{1 + \cos (x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Add $1$ and $1$ .

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

Step 1.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.5.6.1

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

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

Step 1.5.6.2

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

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

Step 1.5.6.3

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

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

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

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

Step 1.5.6.4.2

Rewrite the expression.

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

Step 1.5.6.5

Evaluate the exponent.

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

Step 1.6

Combine using the product rule for radicals.

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

Step 1.7

Apply the product rule to $\frac{\sqrt{(1 + \cos (x)) \cdot 2}}{2}$ .

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

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Step 1.8.1.1

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

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

Step 1.8.1.2

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

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

Step 1.8.1.3

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

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

Step 1.8.1.4

Cancel the common factor of $2$ .

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Step 1.8.1.4.1

Cancel the common factor.

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

Step 1.8.1.4.2

Rewrite the expression.

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

Step 1.8.1.5

Simplify.

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

Step 1.8.2

Apply the distributive property.

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

Step 1.8.3

Multiply $2$ by $1$ .

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

Step 1.8.4

Move $2$ to the left of $\cos (x)$ .

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

Step 1.8.5

Factor $2$ out of $2 + 2 \cos (x)$ .

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Step 1.8.5.1

Factor $2$ out of $2$ .

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

Step 1.8.5.2

Factor $2$ out of $2 \cdot 1 + 2 \cos (x)$ .

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

Step 1.9

Raise $2$ to the power of $2$ .

$2 \frac{2 (1 + \cos (x))}{4} - 1$

Step 1.10

Cancel the common factor of $2$ .

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Step 1.10.1

Factor $2$ out of $4$ .

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

Step 1.10.2

Cancel the common factor.

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

Step 1.10.3

Rewrite the expression.

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

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

Cancel the common factor.

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

Step 1.11.2

Divide $1 + \cos (x)$ by $1$ .

$1 + \cos (x) - 1$

Step 2

Combine the opposite terms in $1 + \cos (x) - 1$ .

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Step 2.1

Subtract $1$ from $1$ .

$0 + \cos (x)$

Step 2.2

Add $0$ and $\cos (x)$ .

$\cos (x)$