Trigonometry Examples

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

Step 1

Simplify $\frac{\cos (x)}{\sec (x) \sin (x)}$ .

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Step 1.1

Separate fractions.

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

Step 1.2

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

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

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Add $1$ and $1$ .

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

Step 1.5

Combine $\frac{1}{\sin (x)}$ and $\cos^{2} (x)$ .

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

Step 1.6

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

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

Step 1.7

Separate fractions.

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

Step 1.8

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

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

Step 1.9

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

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

Step 2

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

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

Step 3