Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3.2

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

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

Step 4

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

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

Step 5

Rewrite $\frac{\cos^{2} (x)}{\sin (x)}$ as $\cot (x) \cos (x)$ .

$\cot (x) \cos (x)$

Step 6

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

$(\csc^{2} (x) - 1) \sin (x) = \cot (x) \cos (x)$ is an identity