Find the Exact Value (4sin(45)cos(45))/(tan(30)cot(30))

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Trigonometry

$\frac{4 \sin (45) \cos (45)}{\tan (30) \cot (30)}$

Simplify the numerator.

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The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{4 \frac{\sqrt{2}}{2} \cos (45)}{\tan (30) \cot (30)}$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{4 \frac{\sqrt{2}}{2} \frac{\sqrt{2}}{2}}{\tan (30) \cot (30)}$

Combine exponents.

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Combine $4$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{4 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}}{\tan (30) \cot (30)}$

Multiply $\frac{4 \sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{4 \sqrt{2} \sqrt{2}}{2 \cdot 2}}{\tan (30) \cot (30)}$

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

$\frac{\frac{4 ({\sqrt{2}}^{1} \sqrt{2})}{2 \cdot 2}}{\tan (30) \cot (30)}$

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

$\frac{\frac{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2 \cdot 2}}{\tan (30) \cot (30)}$

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

$\frac{\frac{4 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}}{\tan (30) \cot (30)}$

Add $1$ and $1$ .

$\frac{\frac{4 {\sqrt{2}}^{2}}{2 \cdot 2}}{\tan (30) \cot (30)}$

Multiply $2$ by $2$ .

$\frac{\frac{4 {\sqrt{2}}^{2}}{4}}{\tan (30) \cot (30)}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{4 {\sqrt{2}}^{2}}{4}$ by cancelling the common factors.

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Cancel the common factor.

$\frac{\frac{4 {\sqrt{2}}^{2}}{4}}{\tan (30) \cot (30)}$

Rewrite the expression.

$\frac{\frac{{\sqrt{2}}^{2}}{1}}{\tan (30) \cot (30)}$

Divide ${\sqrt{2}}^{2}$ by $1$ .

$\frac{{\sqrt{2}}^{2}}{\tan (30) \cot (30)}$

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

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{{(2^{\frac{1}{2}})}^{2}}{\tan (30) \cot (30)}$

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

$\frac{2^{\frac{1}{2} \cdot 2}}{\tan (30) \cot (30)}$

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

$\frac{2^{\frac{2}{2}}}{\tan (30) \cot (30)}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{\tan (30) \cot (30)}$

Divide $1$ by $1$ .

$\frac{2^{1}}{\tan (30) \cot (30)}$

Evaluate the exponent.

$\frac{2}{\tan (30) \cot (30)}$

Simplify the denominator.

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The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2}{\frac{\sqrt{3}}{3} \cot (30)}$

The exact value of $\cot (30)$ is $\sqrt{3}$ .

$\frac{2}{\frac{\sqrt{3}}{3} \sqrt{3}}$

Combine $\frac{\sqrt{3}}{3}$ and $\sqrt{3}$ .

$\frac{2}{\frac{\sqrt{3} \sqrt{3}}{3}}$

Simplify the numerator.

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Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2}{\frac{{\sqrt{3}}^{1} \sqrt{3}}{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2}{\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}}$

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

$\frac{2}{\frac{{\sqrt{3}}^{1 + 1}}{3}}$

Add $1$ and $1$ .

$\frac{2}{\frac{{\sqrt{3}}^{2}}{3}}$

Simplify by cancelling exponent with radical.

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Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2}{\frac{{(3^{\frac{1}{2}})}^{2}}{3}}$

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

$\frac{2}{\frac{3^{\frac{1}{2} \cdot 2}}{3}}$

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

$\frac{2}{\frac{3^{\frac{2}{2}}}{3}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2}{\frac{3^{\frac{2}{2}}}{3}}$

Divide $1$ by $1$ .

$\frac{2}{\frac{3^{1}}{3}}$

Evaluate the exponent.

$\frac{2}{\frac{3}{3}}$

Simplify the expression.

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Divide $3$ by $3$ .

$\frac{2}{1}$

Divide $2$ by $1$ .

$2$

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