Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Add $1$ and $1$ .

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

Step 3.1.3

Rewrite $- 1 \csc (x)$ as $- \csc (x)$ .

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

Step 3.1.4

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

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

Step 3.2

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

${\csc (x)}^{2} + 0 - 1$

Step 3.3

Add ${\csc (x)}^{2}$ and $0$ .

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

Step 4

Apply pythagorean identity.

$\cot^{2} (x)$

Step 5

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

$(\csc (x) - 1) (\csc (x) + 1) = \cot^{2} (x)$ is an identity