Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{2 \cot (2 x)}{\cos (x) - \sin (x)} (\cos (x) - \sin (x))$ .

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Step 2.1.1.1

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

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Step 2.1.1.1.1

Cancel the common factor.

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

Step 2.1.1.1.2

Rewrite the expression.

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

Step 2.1.1.2

Rewrite $\cot (2 x)$ in terms of sines and cosines.

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Separate fractions.

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

Step 2.1.1.5

Convert from $\frac{\cos (2 x)}{\sin (2 x)}$ to $\cot (2 x)$ .

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

Step 2.1.1.6

Divide $2$ by $1$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\csc (x) + \sec (x)) (\cos (x) - \sin (x))$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

Apply the distributive property.

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

Step 2.2.1.2.2

Apply the distributive property.

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

Step 2.2.1.2.3

Apply the distributive property.

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

Step 2.2.1.3

Simplify and combine like terms.

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Step 2.2.1.3.1

Simplify each term.

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Step 2.2.1.3.1.1

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

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

Step 2.2.1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.3.1.3

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

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Step 2.2.1.3.1.3.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 2.2.1.3.1.3.2

Cancel the common factor.

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

Step 2.2.1.3.1.3.3

Rewrite the expression.

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

Step 2.2.1.3.1.4

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

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Step 2.2.1.3.1.4.1

Cancel the common factor.

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

Step 2.2.1.3.1.4.2

Rewrite the expression.

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

Step 2.2.1.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.3.1.6

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

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

Step 2.2.1.3.2

Add $- 1$ and $1$ .

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

Step 2.2.1.3.3

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

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

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

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

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

Step 2.2.1.4.2

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

$2 \cot (2 x) = \cot (x) - \tan (x)$

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 $2 \cot (2 x)$ .

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Step 3.1.1.1

Rewrite $\cot (2 x)$ in terms of sines and cosines.

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

Step 3.1.1.2

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.3

Multiply both sides of the equation by $\sin (2 x)$ .

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

Step 3.4

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

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

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

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

Step 3.7

Rewrite using the commutative property of multiplication.

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

Step 3.8

Simplify each term.

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Apply the sine double-angle identity.

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

Step 3.8.1.2

Combine exponents.

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Step 3.8.1.2.1

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

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

Step 3.8.1.2.2

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

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

Step 3.8.1.2.3

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

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

Step 3.8.1.2.4

Add $1$ and $1$ .

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

Step 3.8.2

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

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Step 3.8.2.1

Cancel the common factor.

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

Step 3.8.2.2

Divide $2 \cos^{2} (x)$ by $1$ .

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

Step 3.8.3

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

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

Step 3.8.4

Simplify the numerator.

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Step 3.8.4.1

Apply the sine double-angle identity.

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

Step 3.8.4.2

Combine exponents.

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Step 3.8.4.2.1

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

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

Step 3.8.4.2.2

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

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

Step 3.8.4.2.3

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

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

Step 3.8.4.2.4

Add $1$ and $1$ .

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

Step 3.8.5

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

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Step 3.8.5.1

Cancel the common factor.

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

Step 3.8.5.2

Divide $2 \sin^{2} (x)$ by $1$ .

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

Step 3.8.6

Multiply $2$ by $- 1$ .

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

Step 3.9

Move all the expressions to the left side of the equation.

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Step 3.9.1

Subtract $2 \cos^{2} (x)$ from both sides of the equation.

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

Step 3.9.2

Add $2 \sin^{2} (x)$ to both sides of the equation.

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

Step 3.10

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

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

Step 3.11

Simplify the left side.

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Step 3.11.1

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

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Step 3.11.1.1

Simplify with factoring out.

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Step 3.11.1.1.1

Factor $2$ out of $2 (1 - 2 \sin^{2} (x)) - 2 \cos^{2} (x) + 2 \sin^{2} (x)$ .

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Step 3.11.1.1.1.1

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

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

Step 3.11.1.1.1.2

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

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

Step 3.11.1.1.1.3

Factor $2$ out of $2 (1 - 2 \sin^{2} (x)) + 2 (- \cos^{2} (x))$ .

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

Step 3.11.1.1.1.4

Factor $2$ out of $2 (1 - 2 \sin^{2} (x) - \cos^{2} (x)) + 2 (\sin^{2} (x))$ .

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

Step 3.11.1.1.2

Move $- \cos^{2} (x)$ .

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

Step 3.11.1.2

Apply pythagorean identity.

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

Step 3.11.1.3

Simplify by adding terms.

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Step 3.11.1.3.1

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

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

Step 3.11.1.3.2

Add $- \sin^{2} (x)$ and $\sin^{2} (x)$ .

$2 \cdot 0 = 0$

Step 3.11.1.3.3

Multiply $2$ by $0$ .

$0 = 0$

Step 3.12

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:

$(- \infty, \infty)$