Trigonometry Examples

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

Step 1

Remove parentheses.

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

Step 2

Apply pythagorean identity.

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

Step 3

Apply pythagorean identity.

$(\csc (x) - 1 \cdot 1) (\csc (x) + 1)$

Step 4

Multiply $- 1$ by $1$ .

$(\csc (x) - 1) (\csc (x) + 1)$

Step 5

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

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Step 5.1

Apply the distributive property.

$\csc (x) (\csc (x) + 1) - 1 (\csc (x) + 1)$

Step 5.2

Apply the distributive property.

$\csc (x) \csc (x) + \csc (x) \cdot 1 - 1 (\csc (x) + 1)$

Step 5.3

Apply the distributive property.

$\csc (x) \csc (x) + \csc (x) \cdot 1 - 1 \csc (x) - 1 \cdot 1$

Step 6

Simplify terms.

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Step 6.1

Combine the opposite terms in $\csc (x) \csc (x) + \csc (x) \cdot 1 - 1 \csc (x) - 1 \cdot 1$ .

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Step 6.1.1

Reorder the factors in the terms $\csc (x) \cdot 1$ and $- 1 \csc (x)$ .

$\csc (x) \csc (x) + 1 \csc (x) - 1 \csc (x) - 1 \cdot 1$

Step 6.1.2

Subtract $1 \csc (x)$ from $1 \csc (x)$ .

$\csc (x) \csc (x) + 0 - 1 \cdot 1$

Step 6.1.3

Add $\csc (x) \csc (x)$ and $0$ .

$\csc (x) \csc (x) - 1 \cdot 1$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $\csc (x) \csc (x)$ .

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Step 6.2.1.1

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

$\csc^{1} (x) \csc (x) - 1 \cdot 1$

Step 6.2.1.2

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

$\csc^{1} (x) \csc^{1} (x) - 1 \cdot 1$

Step 6.2.1.3

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

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

Step 6.2.1.4

Add $1$ and $1$ .

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

Step 6.2.2

Multiply $- 1$ by $1$ .

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

Step 7

Apply pythagorean identity.

$\cot^{2} (x)$