Trigonometry Examples

$\csc^{2} (x) \tan (x) - \tan (x) = \cot (x)$

Step 1

Start on the left side.

$\csc^{2} (x) \tan (x) - \tan (x)$

Step 2

Simplify the expression.

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Step 2.1

Factor $\tan (x)$ out of $\csc^{2} (x) \tan (x) - \tan (x)$ .

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Step 2.1.1

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

$\tan (x) \csc^{2} (x) - \tan (x)$

Step 2.1.2

Factor $\tan (x)$ out of $- \tan (x)$ .

$\tan (x) \csc^{2} (x) + \tan (x) \cdot - 1$

Step 2.1.3

Factor $\tan (x)$ out of $\tan (x) \csc^{2} (x) + \tan (x) \cdot - 1$ .

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

Step 2.2

Apply pythagorean identity.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine.

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

Step 2.7

Cancel the common factor of $\sin (x)$ and $\sin^{2} (x)$ .

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Step 2.7.1

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

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

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

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

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

Step 2.7.2.2

Cancel the common factor.

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

Step 2.7.2.3

Rewrite the expression.

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

Step 2.8

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 2.8.1

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

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

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 3

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

$\cot (x)$

Step 4

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

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