Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Move all terms containing $\sin (2 x)$ to the left side of the equation.

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Step 5.1

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

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

Step 5.2

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

$0 = 0$

Step 6

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

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$