Trigonometry Examples

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

Step 1

Multiply both sides by $1 - \cos (2 x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\cot (x) (1 - \cos (2 x))$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

Apply the distributive property.

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

Step 2.1.1.2.2

Simplify the expression.

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Step 2.1.1.2.2.1

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

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

Step 2.1.1.2.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.1.2.3

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

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

Step 2.1.1.3

Simplify each term.

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Step 2.1.1.3.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2.1.1.3.2

Separate fractions.

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

Step 2.1.1.3.3

Rewrite $\frac{\cos (2 x)}{\sin (x)}$ as a product.

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

Step 2.1.1.3.4

Write $\cos (2 x)$ as a fraction with denominator $1$ .

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

Step 2.1.1.3.5

Simplify.

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Step 2.1.1.3.5.1

Divide $\cos (2 x)$ by $1$ .

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

Step 2.1.1.3.5.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 2.1.1.3.6

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

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $1 - \cos (2 x)$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$\cot (x) - \cos (x) \cos (2 x) \csc (x) = \sin (2 x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

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

Step 3.1.1.3.2

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

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

Step 3.2

Multiply both sides of the equation by $\sin (x)$ .

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $\sin (x)$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

Cancel the common factor of $\sin (x)$ .

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Step 3.6.1

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Step 3.6.2

Cancel the common factor.

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

Step 3.6.3

Rewrite the expression.

$\cos (x) - (\cos (2 x) \cos (x)) = \sin (x) \sin (2 x)$

$\cos (x) - \cos (2 x) \cos (x) = \sin (x) \sin (2 x)$

Step 3.7

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 3.8

Subtract $\sin (x) \sin (2 x)$ from both sides of the equation.

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

Step 3.9

Simplify the left side.

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Step 3.9.1

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

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Step 3.9.1.1

Simplify each term.

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Step 3.9.1.1.1

Apply the distributive property.

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

Step 3.9.1.1.2

Multiply $- 1$ by $1$ .

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

Step 3.9.1.1.3

Multiply $- 2$ by $- 1$ .

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

Step 3.9.1.1.4

Apply the distributive property.

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

Step 3.9.1.1.5

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

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

Step 3.9.1.1.6

Apply the sine double-angle identity.

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

Step 3.9.1.1.7

Multiply $- \sin (x) (2 \sin (x) \cos (x))$ .

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Step 3.9.1.1.7.1

Multiply $2$ by $- 1$ .

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

Step 3.9.1.1.7.2

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

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

Step 3.9.1.1.7.3

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

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

Step 3.9.1.1.7.4

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

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

Step 3.9.1.1.7.5

Add $1$ and $1$ .

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

Step 3.9.1.2

Combine the opposite terms in $\cos (x) - \cos (x) + 2 \sin^{2} (x) \cos (x) - 2 \sin^{2} (x) \cos (x)$ .

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Step 3.9.1.2.1

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

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

Step 3.9.1.2.2

Subtract $2 \sin^{2} (x) \cos (x)$ from $2 \sin^{2} (x) \cos (x)$ .

$0 + 0 = 0$

Step 3.9.1.2.3

Add $0$ and $0$ .

$0 = 0$

Step 3.10

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$