Trigonometry Examples

$(\cot (x) - 1) (\sin (x) - 1) = 0$

Step 1

Simplify $(\cot (x) - 1) (\sin (x) - 1)$ .

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Step 1.1

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

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

Step 1.2

Expand $(\frac{\cos (x)}{\sin (x)} - 1) (\sin (x) - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Cancel the common factor.

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

Step 1.3.1.2

Rewrite the expression.

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

Step 1.3.2

Combine $\frac{\cos (x)}{\sin (x)}$ and $- 1$ .

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

Step 1.3.3

Move $- 1$ to the left of $\cos (x)$ .

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

Step 1.3.4

Move the negative in front of the fraction.

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

Step 1.3.5

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 1.3.6

Multiply $- 1$ by $- 1$ .

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

Step 1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{4} + 2 π n, \frac{π}{2} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 3