Trigonometry Examples

$\frac{\sin^{2} (45)}{\cos^{2} (45)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(\frac{\sqrt{2}}{2})}^{2}}{\cos^{2} (45)}$

Step 1.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{{\sqrt{2}}^{2}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3

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

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Step 1.3.1

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

$\frac{\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3.2

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

$\frac{\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3.3

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

$\frac{\frac{2^{\frac{2}{2}}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{\frac{2^{\frac{2}{2}}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3.4.2

Rewrite the expression.

$\frac{\frac{2^{1}}{2^{2}}}{\cos^{2} (45)}$

Step 1.3.5

Evaluate the exponent.

$\frac{\frac{2}{2^{2}}}{\cos^{2} (45)}$

Step 1.4

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

$\frac{\frac{2}{4}}{\cos^{2} (45)}$

Step 1.5

Cancel the common factor of $2$ and $4$ .

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Step 1.5.1

Factor $2$ out of $2$ .

$\frac{\frac{2 (1)}{4}}{\cos^{2} (45)}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

$\frac{\frac{2 \cdot 1}{2 \cdot 2}}{\cos^{2} (45)}$

Step 1.5.2.2

Cancel the common factor.

$\frac{\frac{2 \cdot 1}{2 \cdot 2}}{\cos^{2} (45)}$

Step 1.5.2.3

Rewrite the expression.

$\frac{\frac{1}{2}}{\cos^{2} (45)}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

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

Step 2.3.4.2

Rewrite the expression.

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

Step 2.3.5

Evaluate the exponent.

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

Step 2.4

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

$\frac{\frac{1}{2}}{\frac{2}{4}}$

Step 2.5

Cancel the common factor of $2$ and $4$ .

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Step 2.5.1

Factor $2$ out of $2$ .

$\frac{\frac{1}{2}}{\frac{2 (1)}{4}}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $4$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

$\frac{\frac{1}{2}}{\frac{1}{2}}$

Step 3

Cancel the common factor of $\frac{1}{2}$ .

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Step 3.1

Cancel the common factor.

$\frac{\frac{1}{2}}{\frac{1}{2}}$

Step 3.2

Rewrite the expression.

$1$