Trigonometry Examples

$1 + \tan^{2} (30 °) - \csc^{2} (45 °)$

Step 1

Rearrange terms.

$\tan^{2} (30 °) + 1 - \csc^{2} (45 °)$

Step 2

Apply pythagorean identity.

$\sec^{2} (30 °) - \csc^{2} (45 °)$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\sec (30 °)$ is $\frac{2}{\sqrt{3}}$ .

${(\frac{2}{\sqrt{3}})}^{2} - \csc^{2} (45 °)$

Step 3.2

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

${(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2} - \csc^{2} (45 °)$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

${(\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2} - \csc^{2} (45 °)$

Step 3.3.2

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

${(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2} - \csc^{2} (45 °)$

Step 3.3.3

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

${(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2} - \csc^{2} (45 °)$

Step 3.3.4

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

${(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2} - \csc^{2} (45 °)$

Step 3.3.5

Add $1$ and $1$ .

${(\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6

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

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

${(\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6.2

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

${(\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6.3

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

${(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

${(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6.4.2

Rewrite the expression.

${(\frac{2 \sqrt{3}}{3^{1}})}^{2} - \csc^{2} (45 °)$

Step 3.3.6.5

Evaluate the exponent.

${(\frac{2 \sqrt{3}}{3})}^{2} - \csc^{2} (45 °)$

Step 3.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$\frac{{(2 \sqrt{3})}^{2}}{3^{2}} - \csc^{2} (45 °)$

Step 3.4.2

Apply the product rule to $2 \sqrt{3}$ .

$\frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5

Simplify the numerator.

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Step 3.5.1

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

$\frac{4 {\sqrt{3}}^{2}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2

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

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Step 3.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2.2

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

$\frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2.3

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

$\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2.4

Cancel the common factor of $2$ .

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Step 3.5.2.4.1

Cancel the common factor.

$\frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2.4.2

Rewrite the expression.

$\frac{4 \cdot 3^{1}}{3^{2}} - \csc^{2} (45 °)$

Step 3.5.2.5

Evaluate the exponent.

$\frac{4 \cdot 3}{3^{2}} - \csc^{2} (45 °)$

Step 3.6

Raise $3$ to the power of $2$ .

$\frac{4 \cdot 3}{9} - \csc^{2} (45 °)$

Step 3.7

Multiply $4$ by $3$ .

$\frac{12}{9} - \csc^{2} (45 °)$

Step 3.8

Cancel the common factor of $12$ and $9$ .

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Step 3.8.1

Factor $3$ out of $12$ .

$\frac{3 (4)}{9} - \csc^{2} (45 °)$

Step 3.8.2

Cancel the common factors.

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Step 3.8.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 4}{3 \cdot 3} - \csc^{2} (45 °)$

Step 3.8.2.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 \cdot 3} - \csc^{2} (45 °)$

Step 3.8.2.3

Rewrite the expression.

$\frac{4}{3} - \csc^{2} (45 °)$

Step 3.9

The exact value of $\csc (45 °)$ is $\sqrt{2}$ .

$\frac{4}{3} - {\sqrt{2}}^{2}$

Step 3.10

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

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Step 3.10.1

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

$\frac{4}{3} - {(2^{\frac{1}{2}})}^{2}$

Step 3.10.2

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

$\frac{4}{3} - 2^{\frac{1}{2} \cdot 2}$

Step 3.10.3

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

$\frac{4}{3} - 2^{\frac{2}{2}}$

Step 3.10.4

Cancel the common factor of $2$ .

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Step 3.10.4.1

Cancel the common factor.

$\frac{4}{3} - 2^{\frac{2}{2}}$

Step 3.10.4.2

Rewrite the expression.

$\frac{4}{3} - 2^{1}$

Step 3.10.5

Evaluate the exponent.

$\frac{4}{3} - 1 \cdot 2$

Step 3.11

Multiply $- 1$ by $2$ .

$\frac{4}{3} - 2$

Step 4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{3} - 2 \cdot \frac{3}{3}$

Step 5

Combine $- 2$ and $\frac{3}{3}$ .

$\frac{4}{3} + \frac{- 2 \cdot 3}{3}$

Step 6

Combine the numerators over the common denominator.

$\frac{4 - 2 \cdot 3}{3}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 2$ by $3$ .

$\frac{4 - 6}{3}$

Step 7.2

Subtract $6$ from $4$ .

$\frac{- 2}{3}$

Step 8

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$