Trigonometry Examples

$\sin (x) + \sin (x) \cot^{2} (x) = \csc (x)$

Step 1

Simplify $\sin (x) + \sin (x) \cot^{2} (x)$ .

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Step 1.1

Multiply by $1$ .

$\sin (x) \cdot 1 + \sin (x) \cot^{2} (x) = \csc (x)$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

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

$\sin (x) \cdot 1 + \sin (x) (\cot^{2} (x)) = \csc (x)$

Step 1.2.2

Factor $\sin (x)$ out of $\sin (x) \cdot 1 + \sin (x) (\cot^{2} (x))$ .

$\sin (x) \cdot (1 + \cot^{2} (x)) = \csc (x)$

Step 1.3

Apply pythagorean identity.

$\sin (x) \cdot \csc^{2} (x) = \csc (x)$

Step 1.4

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

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

Step 1.5

Simplify the expression.

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Step 1.5.1

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 1.5.2

One to any power is one.

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

Step 1.6

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

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factor.

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

Step 1.6.3

Rewrite the expression.

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

Step 1.7

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

$\csc (x) = \csc (x)$

Step 2

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

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

Step 3