Trigonometry Examples

$\cot (- x) \sin (- x) = \cos (x)$

Step 1

Start on the left side.

$\cot (- x) \sin (- x)$

Step 2

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

$- \cot (x) \sin (- x)$

Step 3

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

$- \cot (x) (- \sin (x))$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

$- \cos (x) \cdot - 1$

Step 5.2

Multiply $- 1$ by $- 1$ .

$1 \cos (x)$

Step 5.3

Multiply $\cos (x)$ by $1$ .

$\cos (x)$

Step 6

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

$\cot (- x) \sin (- x) = \cos (x)$ is an identity