Trigonometry Examples

$(\csc (x) + \cot (x)) (1 - \cos (x)) = \sin (x)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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

Step 2.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.3

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

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.6

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

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Step 2.3.1.6.1

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

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

Step 2.3.1.6.2

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

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

Step 2.3.1.6.3

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

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

Step 2.3.1.6.4

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

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

Step 2.3.1.6.5

Add $1$ and $1$ .

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

Step 2.3.2

Add $- \frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

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

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Step 2.6.1

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\frac{\sin (x) \sin (x)}{\sin (x)}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Multiply by $1$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

$\frac{\sin (x)}{1}$

Step 2.6.2.4

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

$\sin (x)$

Step 3

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

$(\csc (x) + \cot (x)) (1 - \cos (x)) = \sin (x)$ is an identity