Trigonometry Examples

$\sin (2 x) \tan (x) + \cos (2 x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\sin (2 x) \tan (x) + \cos (2 x) - 1 = 0$

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the sine double-angle identity.

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

Step 2.3.2

Combine exponents.

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Add $1$ and $1$ .

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

Step 2.4

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

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

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

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

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.