Calculus Examples

$lim_{x \to (\frac{π}{2})} \tan (x) - \sec (x)$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to {(\frac{π}{2})}^{-}} \tan (x) - \sec (x)$

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $\tan (x) - \sec (x)$ by plugging in $(\frac{π}{2})$ for $x$ .

$\tan ((\frac{π}{2})) - \sec ((\frac{π}{2}))$

Step 2.2

The exact value of $\sec (\frac{π}{2})$ is $\frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$ .

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Step 2.2.1

Split $\frac{π}{2}$ into two angles where the values of the six trigonometric functions are known.

$\tan ((\frac{π}{2})) - \sec (\frac{π}{6} + \frac{π}{3})$

Step 2.2.2

Apply the sum of angles identity.

$\tan ((\frac{π}{2})) - \frac{\sec (\frac{π}{6}) \sec (\frac{π}{3}) \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.3

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \sec (\frac{π}{3}) \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.5

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.6

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.7

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.8

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 2.2.9

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \sec (\frac{π}{3})}$

Step 2.2.10

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11

Simplify $\frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$ .

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Step 2.2.11.1

Simplify each term.

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Step 2.2.11.1.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2

Combine and simplify the denominator.

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Step 2.2.11.1.2.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.5

Add $1$ and $1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6

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

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Step 2.2.11.1.2.6.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.11.1.2.6.4.1

Cancel the common factor.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6.4.2

Rewrite the expression.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.2.6.5

Evaluate the exponent.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.3

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

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Step 2.2.11.1.3.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{2 (2 \sqrt{3})}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.3.2

Multiply $2$ by $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 2.2.11.1.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) \cdot 2}$

Step 2.2.11.1.5

Combine and simplify the denominator.

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Step 2.2.11.1.5.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot 2}$

Step 2.2.11.1.5.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot 2}$

Step 2.2.11.1.5.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot 2}$

Step 2.2.11.1.5.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot 2}$

Step 2.2.11.1.5.5

Add $1$ and $1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \cdot 2}$

Step 2.2.11.1.5.6

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

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Step 2.2.11.1.5.6.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot 2}$

Step 2.2.11.1.5.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot 2}$

Step 2.2.11.1.5.6.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot 2}$

Step 2.2.11.1.5.6.4

Cancel the common factor of $2$ .

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Step 2.2.11.1.5.6.4.1

Cancel the common factor.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot 2}$

Step 2.2.11.1.5.6.4.2

Rewrite the expression.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{1}} \cdot 2}$

Step 2.2.11.1.5.6.5

Evaluate the exponent.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot 2}$

Step 2.2.11.1.6

Multiply $- \frac{2 \sqrt{3}}{3} \cdot 2$ .

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Step 2.2.11.1.6.1

Multiply $2$ by $- 1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3}}$

Step 2.2.11.1.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} + \frac{- 2 (2 \sqrt{3})}{3}}$

Step 2.2.11.1.6.3

Multiply $2$ by $- 2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} + \frac{- 4 \sqrt{3}}{3}}$

Step 2.2.11.1.7

Move the negative in front of the fraction.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{4 \sqrt{3}}{3}}$

Step 2.2.11.2

Combine the numerators over the common denominator.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3} - 4 \sqrt{3}}{3}}$

Step 2.2.11.3

Subtract $4 \sqrt{3}$ from $4 \sqrt{3}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{0}{3}}$

Step 2.2.11.4

Divide $0$ by $3$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 2.2.11.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 2.2.12

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 2.3

Since $\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} \tan (x) - \sec (x)$

Step 4

Evaluate the limits by plugging in the value for the variable.

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Step 4.1

Evaluate the limit of $\tan (x) - \sec (x)$ by plugging in $(\frac{π}{2})$ for $x$ .

$\tan ((\frac{π}{2})) - \sec ((\frac{π}{2}))$

Step 4.2

The exact value of $\sec (\frac{π}{2})$ is $\frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$ .

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Step 4.2.1

Split $\frac{π}{2}$ into two angles where the values of the six trigonometric functions are known.

$\tan ((\frac{π}{2})) - \sec (\frac{π}{6} + \frac{π}{3})$

Step 4.2.2

Apply the sum of angles identity.

$\tan ((\frac{π}{2})) - \frac{\sec (\frac{π}{6}) \sec (\frac{π}{3}) \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.3

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \sec (\frac{π}{3}) \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \csc (\frac{π}{6}) \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.5

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \csc (\frac{π}{3})}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.6

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\csc (\frac{π}{6}) \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.7

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \csc (\frac{π}{3}) - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.8

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \sec (\frac{π}{6}) \sec (\frac{π}{3})}$

Step 4.2.9

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \sec (\frac{π}{3})}$

Step 4.2.10

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11

Simplify $\frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2}{\sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$ .

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Step 4.2.11.1

Simplify each term.

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Step 4.2.11.1.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2

Combine and simplify the denominator.

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Step 4.2.11.1.2.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.5

Add $1$ and $1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6

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

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Step 4.2.11.1.2.6.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.11.1.2.6.4.1

Cancel the common factor.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6.4.2

Rewrite the expression.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3^{1}} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.2.6.5

Evaluate the exponent.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{2 \frac{2 \sqrt{3}}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.3

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

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Step 4.2.11.1.3.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{2 (2 \sqrt{3})}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.3.2

Multiply $2$ by $2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2}{\sqrt{3}} \cdot 2}$

Step 4.2.11.1.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) \cdot 2}$

Step 4.2.11.1.5

Combine and simplify the denominator.

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Step 4.2.11.1.5.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot 2}$

Step 4.2.11.1.5.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot 2}$

Step 4.2.11.1.5.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot 2}$

Step 4.2.11.1.5.4

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot 2}$

Step 4.2.11.1.5.5

Add $1$ and $1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \cdot 2}$

Step 4.2.11.1.5.6

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

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Step 4.2.11.1.5.6.1

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot 2}$

Step 4.2.11.1.5.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot 2}$

Step 4.2.11.1.5.6.3

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot 2}$

Step 4.2.11.1.5.6.4

Cancel the common factor of $2$ .

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Step 4.2.11.1.5.6.4.1

Cancel the common factor.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot 2}$

Step 4.2.11.1.5.6.4.2

Rewrite the expression.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{1}} \cdot 2}$

Step 4.2.11.1.5.6.5

Evaluate the exponent.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot 2}$

Step 4.2.11.1.6

Multiply $- \frac{2 \sqrt{3}}{3} \cdot 2$ .

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Step 4.2.11.1.6.1

Multiply $2$ by $- 1$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3}}$

Step 4.2.11.1.6.2

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

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} + \frac{- 2 (2 \sqrt{3})}{3}}$

Step 4.2.11.1.6.3

Multiply $2$ by $- 2$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} + \frac{- 4 \sqrt{3}}{3}}$

Step 4.2.11.1.7

Move the negative in front of the fraction.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3}}{3} - \frac{4 \sqrt{3}}{3}}$

Step 4.2.11.2

Combine the numerators over the common denominator.

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{4 \sqrt{3} - 4 \sqrt{3}}{3}}$

Step 4.2.11.3

Subtract $4 \sqrt{3}$ from $4 \sqrt{3}$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{\frac{0}{3}}$

Step 4.2.11.4

Divide $0$ by $3$ .

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 4.2.11.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 4.2.12

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$

Step 4.3

Since $\tan ((\frac{π}{2})) - \frac{\frac{2}{\sqrt{3}} \cdot 2 \cdot 2 \frac{2}{\sqrt{3}}}{0}$ is undefined, the limit does not exist.

$Does not exist$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$