Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Rearrange terms.

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

Step 2.6

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Combine.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

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

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

Step 5

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

$\cot (x)$

Step 6

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

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