Trigonometry Examples

$\csc (x) - \sin (x) = \cot (x) \cos (x)$

Step 1

Simplify $\csc (x) - \sin (x)$ .

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Step 1.1

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

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

Step 1.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc (x) - \sin (x) = \cot (x) \cos (x)$

Step 2

Simplify $\cot (x) \cos (x)$ .

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.3

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

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

Step 2.4

Separate fractions.

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

Step 2.5

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

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

Step 2.6

Divide $\cos (x)$ by $1$ .

$\csc (x) - \sin (x) = \cos (x) \cot (x)$

Step 3

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

$x \approx 0, 0.1, \frac{1}{5}, 0.3, - \frac{2}{5}, \frac{2}{5}, \frac{1}{2}$

Step 4