Precalculus Examples

$(\csc (θ) + \cot (θ)) (\csc (θ) - \cot (θ)) = 1$

Step 1

Start on the left side.

$(\csc (θ) + \cot (θ)) (\csc (θ) - \cot (θ))$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\csc (θ)$ in terms of sines and cosines.

$(\frac{1}{\sin (θ)} + \cot (θ)) (\csc (θ) - \cot (θ))$

Step 2.1.2

Rewrite $\cot (θ)$ in terms of sines and cosines.

$(\frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)}) (\csc (θ) - \cot (θ))$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite $\csc (θ)$ in terms of sines and cosines.

$(\frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)}) (\frac{1}{\sin (θ)} - \cot (θ))$

Step 2.2.2

Rewrite $\cot (θ)$ in terms of sines and cosines.

$(\frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)}) (\frac{1}{\sin (θ)} - \frac{\cos (θ)}{\sin (θ)})$

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

$\frac{1}{\sin (θ)} (\frac{1}{\sin (θ)} - \frac{\cos (θ)}{\sin (θ)}) + \frac{\cos (θ)}{\sin (θ)} (\frac{1}{\sin (θ)} - \frac{\cos (θ)}{\sin (θ)})$

Step 2.3.2

Apply the distributive property.

$\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{1}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)}) + \frac{\cos (θ)}{\sin (θ)} (\frac{1}{\sin (θ)} - \frac{\cos (θ)}{\sin (θ)})$

Step 2.3.3

Apply the distributive property.

$\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{1}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)}) + \frac{\cos (θ)}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$

Step 2.4

Combine the opposite terms in $\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{1}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)}) + \frac{\cos (θ)}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$ .

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Step 2.4.1

Reorder the factors in the terms $\frac{1}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$ and $\frac{\cos (θ)}{\sin (θ)} \cdot \frac{1}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} - \frac{1}{\sin (θ)} \cdot \frac{\cos (θ)}{\sin (θ)} + \frac{1}{\sin (θ)} \cdot \frac{\cos (θ)}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$

Step 2.4.2

Add $- \frac{1}{\sin (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$ and $\frac{1}{\sin (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + 0 + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$

Step 2.4.3

Add $\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)}$ and $0$ .

$\frac{1}{\sin (θ)} \cdot \frac{1}{\sin (θ)} + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$

Step 2.5

Simplify each term.

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Step 2.5.1

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

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Step 2.5.1.1

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

$\frac{1}{\sin (θ) \sin (θ)} + \frac{\cos (θ)}{\sin (θ)} (- \frac{\cos (θ)}{\sin (θ)})$

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.1.4

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

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

Step 2.5.1.5

Add $1$ and $1$ .

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

Step 2.5.2

Rewrite using the commutative property of multiplication.

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

Step 2.5.3

Multiply $- \frac{\cos (θ)}{\sin (θ)} \cdot \frac{\cos (θ)}{\sin (θ)}$ .

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Step 2.5.3.1

Multiply $\frac{\cos (θ)}{\sin (θ)}$ by $\frac{\cos (θ)}{\sin (θ)}$ .

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

Step 2.5.3.2

Raise $\cos (θ)$ to the power of $1$ .

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

Step 2.5.3.3

Raise $\cos (θ)$ to the power of $1$ .

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

Step 2.5.3.4

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

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

Step 2.5.3.5

Add $1$ and $1$ .

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

Step 2.5.3.6

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

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

Step 2.5.3.7

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

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

Step 2.5.3.8

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

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

Step 2.5.3.9

Add $1$ and $1$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Apply pythagorean identity.

$\frac{\sin^{2} (θ)}{\sin^{2} (θ)}$

Step 2.8

Cancel the common factor of $\sin^{2} (θ)$ .

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Step 2.8.1

Cancel the common factor.

$\frac{\sin^{2} (θ)}{\sin^{2} (θ)}$

Step 2.8.2

Rewrite the expression.

$1$

Step 3

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

$(\csc (θ) + \cot (θ)) (\csc (θ) - \cot (θ)) = 1$ is an identity