Trigonometry Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

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

Step 1.2

Add $\cot (x)$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

To write $- \frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 2.3

Write each expression with a common denominator of $\sin (x) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Multiply $- \sin (x) \sin (x)$ .

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Step 2.5.1.1.1

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

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

Step 2.5.1.1.2

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

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

Step 2.5.1.1.3

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

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

Step 2.5.1.1.4

Add $1$ and $1$ .

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

Step 2.5.1.2

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

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

Step 2.5.1.3

Apply pythagorean identity.

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

Step 2.5.1.4

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

Cancel the common factors.

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Cancel the common factor.

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

Step 2.5.2.2.3

Rewrite the expression.

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

Step 2.5.3

Move the negative in front of the fraction.

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

Step 2.6

Add $- \frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

$0 = 0$

Step 3

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$