Trigonometry Examples

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

Step 1

Apply the sine half-angle identity.

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Add $1$ and $1$ .

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

Step 4.1.6

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

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

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

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

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

Step 4.1.6.4.2

Rewrite the expression.

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

Step 4.1.6.5

Evaluate the exponent.

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

Step 4.2

Combine using the product rule for radicals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

Combine using the product rule for radicals.

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

Step 10

Reorder factors in $(\pm \frac{\sqrt{(1 - \cos (x)) \cdot 2}}{2}) (\pm \frac{\sqrt{(1 + \cos (x)) \cdot 2}}{2})$ .

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