Trigonometry Examples

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

Step 1

Multiply both sides by $\cot (x) - \tan (x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\tan (2 x) (\cot (x) - \tan (x))$ .

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Step 2.1.1.1

Apply the distributive property.

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

Step 2.1.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $\cot (x) - \tan (x)$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Simplify the numerator.

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Step 3.1.1.4.1

Apply the sine double-angle identity.

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

Step 3.1.1.4.2

Combine exponents.

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Step 3.1.1.4.2.1

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

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

Step 3.1.1.4.2.2

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

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

Step 3.1.1.4.2.3

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

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

Step 3.1.1.4.2.4

Add $1$ and $1$ .

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

Step 3.1.1.5

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

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

Step 3.1.1.6

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

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Step 3.1.1.6.1

Cancel the common factor.

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

Step 3.1.1.6.2

Rewrite the expression.

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

Step 3.1.1.7

Apply the cosine double-angle identity.

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

Step 3.1.1.8

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

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

Step 3.1.1.9

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

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

Step 3.1.1.10

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

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

Step 3.1.1.11

Simplify the numerator.

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Step 3.1.1.11.1

Apply the sine double-angle identity.

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

Step 3.1.1.11.2

Combine exponents.

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Step 3.1.1.11.2.1

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

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

Step 3.1.1.11.2.2

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

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

Step 3.1.1.11.2.3

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

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

Step 3.1.1.11.2.4

Add $1$ and $1$ .

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

Step 3.1.1.12

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

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

Step 3.1.1.13

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

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Step 3.1.1.13.1

Cancel the common factor.

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

Step 3.1.1.13.2

Rewrite the expression.

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

Step 3.1.1.14

Apply the cosine double-angle identity.

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

Step 3.2

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

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

Step 3.3

Apply the distributive property.

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

Step 3.4

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

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

Simplify each term.

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Step 3.6.1

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

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Step 3.6.1.1

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

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

Step 3.6.1.2

Cancel the common factor.

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

Step 3.6.1.3

Rewrite the expression.

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

Step 3.6.2

Multiply $2$ by $- 1$ .

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

Step 3.7

Move $2$ to the left of $\cos (2 x)$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Simplify the left side.

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Step 3.10.1

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

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Step 3.10.1.1

Simplify terms.

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Step 3.10.1.1.1

Simplify each term.

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Step 3.10.1.1.1.1

Apply the distributive property.

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

Step 3.10.1.1.1.2

Multiply $- 2$ by $1$ .

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

Step 3.10.1.1.1.3

Multiply $- 2$ by $- 2$ .

$2 \cos^{2} (x) - 2 \sin^{2} (x) - 2 + 4 \sin^{2} (x) = 0$

Step 3.10.1.1.2

Simplify with factoring out.

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Step 3.10.1.1.2.1

Move $- 2$ .

$2 \cos^{2} (x) - 2 - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.1.2.2

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

$- 2 + 2 \cos^{2} (x) - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.1.2.3

Factor $- 2$ out of $- 2$ .

$- 2 (1) + 2 \cos^{2} (x) - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.1.2.4

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

$- 2 (1) - 2 (- \cos^{2} (x)) - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.1.2.5

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

$- 2 (1 - \cos^{2} (x)) - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.2

Apply pythagorean identity.

$- 2 \sin^{2} (x) - 2 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.3

Simplify by adding terms.

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Step 3.10.1.3.1

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

$- 4 \sin^{2} (x) + 4 \sin^{2} (x) = 0$

Step 3.10.1.3.2

Add $- 4 \sin^{2} (x)$ and $4 \sin^{2} (x)$ .

$0 = 0$

Step 3.11

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:

$(- \infty, \infty)$