Trigonometry Examples

$\tan (- x) \cot (x) = - 1$

Step 1

Start on the left side.

$\tan (- x) \cot (x)$

Step 2

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

$- \tan (x) \cot (x)$

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.

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

Move the leading negative in $- \frac{\sin (x)}{\cos (x)}$ into the numerator.

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

Step 4.1.2

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

$\frac{\sin (x) \cdot - 1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 4.1.3

Cancel the common factor.

$\frac{\sin (x) \cdot - 1}{\cos (x)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 4.1.4

Rewrite the expression.

$\frac{- 1}{\cos (x)} \cos (x)$

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

$\frac{- 1}{\cos (x)} \cos (x)$

Step 4.2.2

Rewrite the expression.

$- 1$

Step 5

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

$\tan (- x) \cot (x) = - 1$ is an identity