Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

Multiply by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

One to any power is one.

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

Step 4.2

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

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

Step 5

Rewrite $\frac{1}{\sin (x)}$ as $\csc (x)$ .

$\csc (x)$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

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