Algebra Examples

$\csc^{2} (x) - \sec^{2} (x) = \cot^{2} (x) - \tan^{2} (x)$

Step 1

Start on the right side.

$\cot^{2} (x) - \tan^{2} (x)$

Step 2

Apply Pythagorean identity in reverse.

$\csc^{2} (x) - 1 - \tan^{2} (x)$

Step 3

Factor.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\csc^{2} (x) - 1^{2} - \tan^{2} (x)$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \csc (x)$ and $b = 1$ .

$(\csc (x) + 1) (\csc (x) - 1) - \tan^{2} (x)$

Step 4

Apply Pythagorean identity in reverse.

$(\csc (x) + 1) (\csc (x) - 1) - (\sec^{2} (x) - 1)$

Step 5

Convert to sines and cosines.

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Step 5.1

Apply the reciprocal identity to $\csc (x)$ .

$(\frac{1}{\sin (x)} + 1) (\csc (x) - 1) - (\sec^{2} (x) - 1)$

Step 5.2

Apply the reciprocal identity to $\csc (x)$ .

$(\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1) - (\sec^{2} (x) - 1)$

Step 5.3

Apply the reciprocal identity to $\sec (x)$ .

$(\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1) - ({(\frac{1}{\cos (x)})}^{2} - 1)$

Step 5.4

Apply the product rule to $\frac{1}{\cos (x)}$ .

$(\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1) - (\frac{1^{2}}{\cos^{2} (x)} - 1)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Expand $(\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)$ using the FOIL Method.

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Step 6.1.1.1

Apply the distributive property.

$\frac{1}{\sin (x)} (\frac{1}{\sin (x)} - 1) + 1 (\frac{1}{\sin (x)} - 1) - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.1.2

Apply the distributive property.

$\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{1}{\sin (x)} \cdot - 1 + 1 (\frac{1}{\sin (x)} - 1) - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.1.3

Apply the distributive property.

$\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2

Simplify and combine like terms.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Step 6.1.2.1.1.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\sin (x)}$ .

$\frac{1}{\sin (x) \sin (x)} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.1.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{1} \sin (x)} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.1.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{1} {\sin (x)}^{1}} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.1.4

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

$\frac{1}{{\sin (x)}^{1 + 1}} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.1.5

Add $1$ and $1$ .

$\frac{1}{{\sin (x)}^{2}} + \frac{1}{\sin (x)} \cdot - 1 + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.2

Combine $\frac{1}{\sin (x)}$ and $- 1$ .

$\frac{1}{{\sin (x)}^{2}} + \frac{- 1}{\sin (x)} + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.3

Move the negative in front of the fraction.

$\frac{1}{{\sin (x)}^{2}} - \frac{1}{\sin (x)} + 1 \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.4

Multiply $\frac{1}{\sin (x)}$ by $1$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{1}{\sin (x)} + \frac{1}{\sin (x)} + 1 \cdot - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.1.5

Multiply $- 1$ by $1$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{1}{\sin (x)} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.2

To write $- \frac{1}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{1}{\sin (x)} \cdot \frac{\sin (x)}{\sin (x)} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.3

Write each expression with a common denominator of ${\sin (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.2.3.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{\sin (x)}{\sin (x) \sin (x)} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.3.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{\sin (x)}{{\sin (x)}^{1} \sin (x)} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.3.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{\sin (x)}{{\sin (x)}^{1} {\sin (x)}^{1}} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.3.4

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

$\frac{1}{{\sin (x)}^{2}} - \frac{\sin (x)}{{\sin (x)}^{1 + 1}} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.3.5

Add $1$ and $1$ .

$\frac{1}{{\sin (x)}^{2}} - \frac{\sin (x)}{{\sin (x)}^{2}} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.4

Combine the numerators over the common denominator.

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{1}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.5

To write $\frac{1}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{1}{\sin (x)} \cdot \frac{\sin (x)}{\sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.6

Write each expression with a common denominator of ${\sin (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.2.6.1

Multiply $\frac{1}{\sin (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{\sin (x)}{\sin (x) \sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.6.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{\sin (x)}{{\sin (x)}^{1} \sin (x)} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.6.3

Raise $\sin (x)$ to the power of $1$ .

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{\sin (x)}{{\sin (x)}^{1} {\sin (x)}^{1}} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.6.4

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

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{\sin (x)}{{\sin (x)}^{1 + 1}} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.6.5

Add $1$ and $1$ .

$\frac{1 - \sin (x)}{{\sin (x)}^{2}} + \frac{\sin (x)}{{\sin (x)}^{2}} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.7

Combine the numerators over the common denominator.

$\frac{1 - \sin (x) + \sin (x)}{{\sin (x)}^{2}} - 1 - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{1 - \sin (x) + \sin (x)}{{\sin (x)}^{2}} - 1 \cdot \frac{{\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.9

Combine $- 1$ and $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{1 - \sin (x) + \sin (x)}{{\sin (x)}^{2}} + \frac{- {\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.2.10

Combine the numerators over the common denominator.

$\frac{1 - \sin (x) + \sin (x) - {\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Add $- \sin (x)$ and $\sin (x)$ .

$\frac{1 + 0 - {\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.3.2

Add $1$ and $0$ .

$\frac{1 - {\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.3.3

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - {\sin (x)}^{2}}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} - (\frac{1^{2}}{{\cos (x)}^{2}} - 1)$

Step 6.1.4

One to any power is one.

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} - (\frac{1}{{\cos (x)}^{2}} - 1)$

Step 6.1.5

Apply the distributive property.

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} - \frac{1}{{\cos (x)}^{2}} - - 1$

Step 6.1.6

Multiply $- 1$ by $- 1$ .

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} - \frac{1}{{\cos (x)}^{2}} + 1$

Step 6.2

To write $\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} \cdot \frac{{\cos (x)}^{2}}{{\cos (x)}^{2}} - \frac{1}{{\cos (x)}^{2}} + 1$

Step 6.3

To write $- \frac{1}{{\cos (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}} \cdot \frac{{\cos (x)}^{2}}{{\cos (x)}^{2}} - \frac{1}{{\cos (x)}^{2}} \cdot \frac{{\sin (x)}^{2}}{{\sin (x)}^{2}} + 1$

Step 6.4

Write each expression with a common denominator of ${\sin (x)}^{2} {\cos (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.1

Multiply $\frac{(1 + \sin (x)) (1 - \sin (x))}{{\sin (x)}^{2}}$ by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

$\frac{(1 + \sin (x)) (1 - \sin (x)) {\cos (x)}^{2}}{{\sin (x)}^{2} {\cos (x)}^{2}} - \frac{1}{{\cos (x)}^{2}} \cdot \frac{{\sin (x)}^{2}}{{\sin (x)}^{2}} + 1$

Step 6.4.2

Multiply $\frac{1}{{\cos (x)}^{2}}$ by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{(1 + \sin (x)) (1 - \sin (x)) {\cos (x)}^{2}}{{\sin (x)}^{2} {\cos (x)}^{2}} - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

$\frac{(1 + \sin (x)) (1 - \sin (x)) {\cos (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} - \frac{{\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.5

Combine the numerators over the common denominator.

$\frac{(1 + \sin (x)) (1 - \sin (x)) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Expand $(1 + \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Step 6.6.1.1

Apply the distributive property.

$\frac{(1 (1 - \sin (x)) + \sin (x) (1 - \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.1.2

Apply the distributive property.

$\frac{(1 \cdot 1 + 1 (- \sin (x)) + \sin (x) (1 - \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.1.3

Apply the distributive property.

$\frac{(1 \cdot 1 + 1 (- \sin (x)) + \sin (x) \cdot 1 + \sin (x) (- \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2

Simplify and combine like terms.

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Step 6.6.2.1

Simplify each term.

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Step 6.6.2.1.1

Multiply $1$ by $1$ .

$\frac{(1 + 1 (- \sin (x)) + \sin (x) \cdot 1 + \sin (x) (- \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.2

Multiply $- \sin (x)$ by $1$ .

$\frac{(1 - \sin (x) + \sin (x) \cdot 1 + \sin (x) (- \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.3

Multiply $\sin (x)$ by $1$ .

$\frac{(1 - \sin (x) + \sin (x) + \sin (x) (- \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.4

Multiply $\sin (x) (- \sin (x))$ .

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Step 6.6.2.1.4.1

Raise $\sin (x)$ to the power of $1$ .

$\frac{(1 - \sin (x) + \sin (x) - ({\sin (x)}^{1} \sin (x))) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.4.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{(1 - \sin (x) + \sin (x) - ({\sin (x)}^{1} {\sin (x)}^{1})) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.4.3

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

$\frac{(1 - \sin (x) + \sin (x) - {\sin (x)}^{1 + 1}) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.1.4.4

Add $1$ and $1$ .

$\frac{(1 - \sin (x) + \sin (x) - {\sin (x)}^{2}) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.2

Add $- \sin (x)$ and $\sin (x)$ .

$\frac{(1 + 0 - {\sin (x)}^{2}) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.2.3

Add $1$ and $0$ .

$\frac{(1 - {\sin (x)}^{2}) {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.3

Apply the distributive property.

$\frac{1 {\cos (x)}^{2} - {\sin (x)}^{2} {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.6.4

Multiply ${\cos (x)}^{2}$ by $1$ .

$\frac{{\cos (x)}^{2} - {\sin (x)}^{2} {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + 1$

Step 6.7

Write $1$ as a fraction with a common denominator.

$\frac{{\cos (x)}^{2} - {\sin (x)}^{2} {\cos (x)}^{2} - {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}} + \frac{{\cos (x)}^{2} {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}}$

Step 6.8

Combine the numerators over the common denominator.

$\frac{{\cos (x)}^{2} - {\sin (x)}^{2} {\cos (x)}^{2} - {\sin (x)}^{2} + {\cos (x)}^{2} {\sin (x)}^{2}}{{\cos (x)}^{2} {\sin (x)}^{2}}$

Step 6.9

Simplify the numerator.

$\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{\cos^{2} (x) \sin^{2} (x)}$

Step 7

Now consider the left side of the equation.

$\csc^{2} (x) - \sec^{2} (x)$

Step 8

Convert to sines and cosines.

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Step 8.1

Apply the reciprocal identity to $\csc (x)$ .

${(\frac{1}{\sin (x)})}^{2} - \sec^{2} (x)$

Step 8.2

Apply the reciprocal identity to $\sec (x)$ .

${(\frac{1}{\sin (x)})}^{2} - {(\frac{1}{\cos (x)})}^{2}$

Step 8.3

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{1^{2}}{\sin^{2} (x)} - {(\frac{1}{\cos (x)})}^{2}$

Step 8.4

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{1^{2}}{\sin^{2} (x)} - \frac{1^{2}}{\cos^{2} (x)}$

Step 9

Simplify each term.

$\frac{1}{\sin^{2} (x)} - \frac{1}{\cos^{2} (x)}$

Step 10

Subtract fractions.

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Step 10.1

To write $\frac{1}{\sin^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{1}{\sin^{2} (x)} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)}$

Step 10.2

To write $- \frac{1}{\cos^{2} (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$\frac{1}{\sin^{2} (x)} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)}$

Step 10.3

Write each expression with a common denominator of $\sin^{2} (x) \cos^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.1

Multiply $\frac{1}{\sin^{2} (x)}$ by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{\cos^{2} (x)}{\sin^{2} (x) \cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\sin^{2} (x)}$

Step 10.3.2

Multiply $\frac{1}{\cos^{2} (x)}$ by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

$\frac{\cos^{2} (x)}{\sin^{2} (x) \cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

Step 10.3.3

Reorder the factors of $\sin^{2} (x) \cos^{2} (x)$ .

$\frac{\cos^{2} (x)}{\cos^{2} (x) \sin^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

Step 10.4

Combine the numerators over the common denominator.

$\frac{\cos^{2} (x) - \sin^{2} (x)}{\cos^{2} (x) \sin^{2} (x)}$

Step 11

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

$\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{\cos^{2} (x) \sin^{2} (x)}$

Step 12

Because the two sides have been shown to be equivalent, the equation is an identity.

$\csc^{2} (x) - \sec^{2} (x) = \cot^{2} (x) - \tan^{2} (x)$ is an identity