Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\csc (t) \csc (t)$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

Multiply $- 1$ by $1$ .

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

Step 3

Apply pythagorean identity.

$\cot^{2} (t)$