Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

To write $\frac{1}{1 - \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \cos (x)}{1 + \cos (x)}$ .

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

Step 1.1.2

To write $\frac{1}{1 + \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cos (x)}{1 - \cos (x)}$ .

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

Step 1.1.3

Write each expression with a common denominator of $(1 - \cos (x)) (1 + \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Reorder the factors of $(1 - \cos (x)) (1 + \cos (x))$ .

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Add $1$ and $1$ .

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

Step 1.1.5.2

Subtract $\cos (x)$ from $\cos (x)$ .

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

Step 1.1.5.3

Add $2$ and $0$ .

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

Step 2

Multiply both sides by $(1 + \cos (x)) (1 - \cos (x))$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $(1 + \cos (x)) (1 - \cos (x))$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

Combine fractions.

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Step 3.2.1.2.1

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 3.2.1.2.2

One to any power is one.

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

Step 3.2.1.2.3

Combine $2$ and $\frac{1}{\sin^{2} (x)}$ .

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

Apply the distributive property.

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

Step 3.2.1.3.2

Apply the distributive property.

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

Step 3.2.1.3.3

Apply the distributive property.

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

Step 3.2.1.4

Simplify and combine like terms.

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Step 3.2.1.4.1

Simplify each term.

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Step 3.2.1.4.1.1

Multiply $1$ by $1$ .

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

Step 3.2.1.4.1.2

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

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

Step 3.2.1.4.1.3

Multiply $\cos (x)$ by $1$ .

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

Step 3.2.1.4.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.4.1.5

Multiply $- \cos (x) \cos (x)$ .

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Step 3.2.1.4.1.5.1

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

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

Step 3.2.1.4.1.5.2

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

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

Step 3.2.1.4.1.5.3

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

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

Step 3.2.1.4.1.5.4

Add $1$ and $1$ .

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

Step 3.2.1.4.2

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

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

Step 3.2.1.4.3

Add $1$ and $0$ .

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

Step 3.2.1.5

Apply pythagorean identity.

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

Step 3.2.1.6

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

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Step 3.2.1.6.1

Cancel the common factor.

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

Step 3.2.1.6.2

Rewrite the expression.

$2 = 2$

Step 4

Since $2 = 2$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$