Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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

Step 9

Simplify the numerator.

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Step 9.1

Apply the sine double-angle identity.

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

Step 9.2

Combine exponents.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

Add $1$ and $1$ .

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

Step 10

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

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Step 10.1

Cancel the common factor.

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

Step 10.2

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

Apply the cosine double-angle identity.

$\cos (2 x) - \cos (2 x) = 0$

Step 12.3

Subtract $\cos (2 x)$ from $\cos (2 x)$ .

$0 = 0$

Step 13

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

All real numbers

Step 14

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$