Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify $\frac{2 \tan (x)}{\sin (2 x)} - \sec^{2} (x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Separate fractions.

$\frac{2}{1} \cdot \frac{\tan (x)}{\sin (2 x)} - \sec^{2} (x) = 0$

Step 2.1.2

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

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

Step 2.1.3

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\sin (2 x)}$ as a product.

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

Step 2.1.4

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\sin (2 x)}$ .

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

Step 2.1.5

Divide $2$ by $1$ .

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

Step 2.1.6

Simplify the denominator.

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Step 2.1.6.1

Apply the sine double-angle identity.

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

Step 2.1.6.2

Combine exponents.

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Step 2.1.6.2.1

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

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

Step 2.1.6.2.2

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

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

Step 2.1.6.2.3

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

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

Step 2.1.6.2.4

Add $1$ and $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

Cancel the common factor.

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

Step 2.1.7.2

Rewrite the expression.

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

Step 2.1.8

Cancel the common factor of $2$ .

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Step 2.1.8.1

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

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

Step 2.1.8.2

Cancel the common factor.

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

Step 2.1.8.3

Rewrite the expression.

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

Step 2.1.9

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

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

Step 2.1.10

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.1.11

One to any power is one.

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

Step 2.2

Subtract $\frac{1}{\cos^{2} (x)}$ from $\frac{1}{\cos^{2} (x)}$ .

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.