Trigonometry Examples

$\frac{1}{\frac{121}{1}} - \frac{1}{\frac{y^{2}}{1}}$

Step 1

Simplify each term.

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Step 1.1

Divide $121$ by $1$ .

$\frac{1}{121} - \frac{1}{\frac{y^{2}}{1}}$

Step 1.2

Divide $y^{2}$ by $1$ .

$\frac{1}{121} - \frac{1}{y^{2}}$

Step 2

To write $\frac{1}{121}$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

$\frac{1}{121} \cdot \frac{y^{2}}{y^{2}} - \frac{1}{y^{2}}$

Step 3

To write $- \frac{1}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{121}{121}$ .

$\frac{1}{121} \cdot \frac{y^{2}}{y^{2}} - \frac{1}{y^{2}} \cdot \frac{121}{121}$

Step 4

Write each expression with a common denominator of $121 y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{121}$ by $\frac{y^{2}}{y^{2}}$ .

$\frac{y^{2}}{121 y^{2}} - \frac{1}{y^{2}} \cdot \frac{121}{121}$

Step 4.2

Multiply $\frac{1}{y^{2}}$ by $\frac{121}{121}$ .

$\frac{y^{2}}{121 y^{2}} - \frac{121}{y^{2} \cdot 121}$

Step 4.3

Reorder the factors of $y^{2} \cdot 121$ .

$\frac{y^{2}}{121 y^{2}} - \frac{121}{121 y^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{y^{2} - 121}{121 y^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $121$ as $11^{2}$ .

$\frac{y^{2} - 11^{2}}{121 y^{2}}$

Step 6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 11$ .

$\frac{(y + 11) (y - 11)}{121 y^{2}}$