Trigonometry Examples

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

Step 1

Multiply both sides by $1 - \tan^{2} (x)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Apply the distributive property.

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

Step 2.1.1.2

Simplify the expression.

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Step 2.1.1.2.1

Multiply $\tan (2 x)$ by $1$ .

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

Step 2.1.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $1 - \tan^{2} (x)$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

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 each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

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

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

Step 3.1.1.6

Simplify the numerator.

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Step 3.1.1.6.1

Apply the sine double-angle identity.

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

Step 3.1.1.6.2

Combine exponents.

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Step 3.1.1.6.2.1

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

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

Step 3.1.1.6.2.2

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

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

Step 3.1.1.6.2.3

Add $2$ and $1$ .

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x) \cos (x)}{\cos^{2} (x) \cos (2 x)} = 2 \tan (x)$

Step 3.1.1.7

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

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x) \cos (x)}{\cos^{2} (x) (2 \cos^{2} (x) - 1)} = 2 \tan (x)$

Step 3.1.1.8

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

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Step 3.1.1.8.1

Factor $\cos (x)$ out of $2 \sin^{3} (x) \cos (x)$ .

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{\cos (x) (2 \sin^{3} (x))}{\cos^{2} (x) (2 \cos^{2} (x) - 1)} = 2 \tan (x)$

Step 3.1.1.8.2

Cancel the common factors.

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Step 3.1.1.8.2.1

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

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{\cos (x) (2 \sin^{3} (x))}{\cos (x) (\cos (x) (2 \cos^{2} (x) - 1))} = 2 \tan (x)$

Step 3.1.1.8.2.2

Cancel the common factor.

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{\cos (x) (2 \sin^{3} (x))}{\cos (x) (\cos (x) (2 \cos^{2} (x) - 1))} = 2 \tan (x)$

Step 3.1.1.8.2.3

Rewrite the expression.

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x)}{\cos (x) (2 \cos^{2} (x) - 1)} = 2 \tan (x)$

Step 3.1.1.9

Apply the cosine double-angle identity.

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)} = 2 \tan (x)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $2 \tan (x)$ .

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Step 3.2.1.1

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

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)} = 2 \frac{\sin (x)}{\cos (x)}$

Step 3.2.1.2

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

$\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)} = \frac{2 \sin (x)}{\cos (x)}$

Step 3.3

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

$\cos (2 x) (\frac{\sin (2 x)}{\cos (2 x)} - \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)}) = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.4

Apply the distributive property.

$\cos (2 x) \frac{\sin (2 x)}{\cos (2 x)} + \cos (2 x) (- \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)}) = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.5

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

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Step 3.5.1

Cancel the common factor.

$\cos (2 x) \frac{\sin (2 x)}{\cos (2 x)} + \cos (2 x) (- \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)}) = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.5.2

Rewrite the expression.

$\sin (2 x) + \cos (2 x) (- \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)}) = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.6

Rewrite using the commutative property of multiplication.

$\sin (2 x) - \cos (2 x) \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)} = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.7

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

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Step 3.7.1

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

$\sin (2 x) + \cos (2 x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (x) \cos (2 x)} = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.7.2

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

$\sin (2 x) + \cos (2 x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (2 x) \cos (x)} = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.7.3

Cancel the common factor.

$\sin (2 x) + \cos (2 x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (2 x) \cos (x)} = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.7.4

Rewrite the expression.

$\sin (2 x) - \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (2 x) \frac{2 \sin (x)}{\cos (x)}$

Step 3.8

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

$\sin (2 x) - \frac{2 \sin^{3} (x)}{\cos (x)} = \frac{\cos (2 x) (2 \sin (x))}{\cos (x)}$

Step 3.9

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

$\sin (2 x) - \frac{2 \sin^{3} (x)}{\cos (x)} = \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.10

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

$\cos (x) (\sin (2 x) - \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.11

Apply the distributive property.

$\cos (x) \sin (2 x) + \cos (x) (- \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.12

Rewrite using the commutative property of multiplication.

$\cos (x) \sin (2 x) - \cos (x) \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.13

Simplify each term.

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Step 3.13.1

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

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Step 3.13.1.1

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

$\cos (x) \sin (2 x) + \cos (x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.13.1.2

Cancel the common factor.

$\cos (x) \sin (2 x) + \cos (x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.13.1.3

Rewrite the expression.

$\cos (x) \sin (2 x) - (2 \sin^{3} (x)) = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.13.2

Multiply $2$ by $- 1$ .

$\cos (x) \sin (2 x) - 2 \sin^{3} (x) = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.14

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

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Step 3.14.1

Cancel the common factor.

$\cos (x) \sin (2 x) - 2 \sin^{3} (x) = \cos (x) \frac{2 \cos (2 x) \sin (x)}{\cos (x)}$

Step 3.14.2

Rewrite the expression.

$\cos (x) \sin (2 x) - 2 \sin^{3} (x) = 2 \cos (2 x) \sin (x)$

Step 3.15

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

$\cos (x) \sin (2 x) - 2 \sin^{3} (x) = 2 (1 - 2 \sin^{2} (x)) \sin (x)$

Step 3.16

Subtract $2 (1 - 2 \sin^{2} (x)) \sin (x)$ from both sides of the equation.

$\cos (x) \sin (2 x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17

Simplify the left side.

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Step 3.17.1

Simplify $\cos (x) \sin (2 x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x)$ .

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Step 3.17.1.1

Simplify terms.

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Step 3.17.1.1.1

Simplify each term.

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Step 3.17.1.1.1.1

Apply the sine double-angle identity.

$\cos (x) (2 \sin (x) \cos (x)) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.2

Rewrite using the commutative property of multiplication.

$2 \cos (x) \sin (x) \cos (x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.3

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

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Step 3.17.1.1.1.3.1

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

$2 (\cos^{1} (x) \cos (x)) \sin (x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.3.2

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

$2 (\cos^{1} (x) \cos^{1} (x)) \sin (x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.3.3

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

$2 {\cos (x)}^{1 + 1} \sin (x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.3.4

Add $1$ and $1$ .

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 (1 - 2 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.4

Apply the distributive property.

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) + (- 2 \cdot 1 - 2 (- 2 \sin^{2} (x))) \sin (x) = 0$

Step 3.17.1.1.1.5

Multiply $- 2$ by $1$ .

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) + (- 2 - 2 (- 2 \sin^{2} (x))) \sin (x) = 0$

Step 3.17.1.1.1.6

Multiply $- 2$ by $- 2$ .

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) + (- 2 + 4 \sin^{2} (x)) \sin (x) = 0$

Step 3.17.1.1.1.7

Apply the distributive property.

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 \sin^{2} (x) \sin (x) = 0$

Step 3.17.1.1.1.8

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 3.17.1.1.1.8.1

Move $\sin (x)$ .

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 (\sin (x) \sin^{2} (x)) = 0$

Step 3.17.1.1.1.8.2

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

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Step 3.17.1.1.1.8.2.1

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

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 (\sin^{1} (x) \sin^{2} (x)) = 0$

Step 3.17.1.1.1.8.2.2

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

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 {\sin (x)}^{1 + 2} = 0$

Step 3.17.1.1.1.8.3

Add $1$ and $2$ .

$2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 \sin^{3} (x) = 0$

Step 3.17.1.1.2

Simplify with factoring out.

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Step 3.17.1.1.2.1

Factor $2 \sin (x)$ out of $2 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x) - 2 \sin (x) + 4 \sin^{3} (x)$ .

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Step 3.17.1.1.2.1.1

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

$2 \sin (x) (\cos^{2} (x)) - 2 \sin^{3} (x) - 2 \sin (x) + 4 \sin^{3} (x) = 0$

Step 3.17.1.1.2.1.2

Factor $2 \sin (x)$ out of $- 2 \sin^{3} (x)$ .

$2 \sin (x) (\cos^{2} (x)) + 2 \sin (x) (- \sin^{2} (x)) - 2 \sin (x) + 4 \sin^{3} (x) = 0$

Step 3.17.1.1.2.1.3

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

$2 \sin (x) (\cos^{2} (x)) + 2 \sin (x) (- \sin^{2} (x)) + 2 \sin (x) (- 1) + 4 \sin^{3} (x) = 0$

Step 3.17.1.1.2.1.4

Factor $2 \sin (x)$ out of $4 \sin^{3} (x)$ .

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

Step 3.17.1.1.2.1.5

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

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

Step 3.17.1.1.2.1.6

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

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

Step 3.17.1.1.2.1.7

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

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

Step 3.17.1.1.2.2

Simplify the expression.

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Step 3.17.1.1.2.2.1

Move $- 1$ .

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

Step 3.17.1.1.2.2.2

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 3.17.1.1.2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 3.17.1.1.2.4

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

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

Step 3.17.1.1.2.5

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

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

Step 3.17.1.1.2.6

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 3.17.1.2

Apply pythagorean identity.

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

Step 3.17.1.3

Simplify by adding terms.

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Step 3.17.1.3.1

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

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

Step 3.17.1.3.2

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

$2 \sin (x) \cdot 0 = 0$

Step 3.17.1.4

Multiply $2 \sin (x) \cdot 0$ .

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Step 3.17.1.4.1

Multiply $0$ by $2$ .

$0 \sin (x) = 0$

Step 3.17.1.4.2

Multiply $0$ by $\sin (x)$ .

$0 = 0$

Step 3.18

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:

$(- \infty, \infty)$