Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

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

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Simplify each term.

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Step 8.1

Apply the sine double-angle identity.

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

Step 8.2

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

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Apply the sine double-angle identity.

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

Step 8.6

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

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Step 8.6.1

Cancel the common factor.

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

Step 8.6.2

Rewrite the expression.

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

Step 8.7

Apply the cosine double-angle identity.

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

Step 9

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

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

Step 10

Move all terms containing $x$ to the left side of the equation.

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Step 10.1

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

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

Step 10.2

Add $\frac{1 - 2 \sin^{2} (x)}{2 \sin (x)}$ to both sides of the equation.

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

Step 11

Simplify the left side.

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Step 11.1

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

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Step 11.1.1

Combine fractions.

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Step 11.1.1.1

Combine the numerators over the common denominator.

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

Step 11.1.1.2

Simplify by adding numbers.

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Step 11.1.1.2.1

Add $- 1$ and $1$ .

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

Step 11.1.1.2.2

Subtract $2 \sin^{2} (x)$ from $0$ .

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

Step 11.1.2

Simplify each term.

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Step 11.1.2.1

Cancel the common factor of $- 2$ and $2$ .

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Step 11.1.2.1.1

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

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

Step 11.1.2.1.2

Cancel the common factors.

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Step 11.1.2.1.2.1

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

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

Step 11.1.2.1.2.2

Cancel the common factor.

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

Step 11.1.2.1.2.3

Rewrite the expression.

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

Step 11.1.2.2

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

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Step 11.1.2.2.1

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

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

Step 11.1.2.2.2

Cancel the common factors.

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Step 11.1.2.2.2.1

Multiply by $1$ .

$\sin (x) + \frac{\sin (x) (- \sin (x))}{\sin (x) \cdot 1} = 0$

Step 11.1.2.2.2.2

Cancel the common factor.

$\sin (x) + \frac{\sin (x) (- \sin (x))}{\sin (x) \cdot 1} = 0$

Step 11.1.2.2.2.3

Rewrite the expression.

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

Step 11.1.2.2.2.4

Divide $- \sin (x)$ by $1$ .

$\sin (x) - \sin (x) = 0$

Step 11.1.3

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

$0 = 0$

Step 12

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

All real numbers

Step 13

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$