Trigonometry Examples

$\sin^{2} (\frac{x}{2})$

Step 1

Apply the sine half-angle identity.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Combine and simplify the denominator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Add $1$ and $1$ .

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

Step 5.1.6

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

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Step 5.1.6.1

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

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

Step 5.1.6.2

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

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

Step 5.1.6.3

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

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

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

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

Step 5.1.6.4.2

Rewrite the expression.

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

Step 5.1.6.5

Evaluate the exponent.

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

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

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Cancel the common factor.

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

Step 6.1.4.2

Rewrite the expression.

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

Step 6.1.5

Simplify.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $2$ by $1$ .

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

Step 6.4

Multiply $2$ by $- 1$ .

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

Step 6.5

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

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Step 6.5.1

Factor $2$ out of $2$ .

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

Step 6.5.2

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

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

Step 6.5.3

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

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

Step 7

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

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

Step 8

Cancel the common factors.

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Step 8.1

Factor $2$ out of $4$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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