Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Dividing two negative values results in a positive value.

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

Step 6.2

Combine $\cos (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 6.3

Reduce the expression by cancelling the common factors.

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Step 6.3.1

Reduce the expression $\frac{\cos (x) \sin (x)}{\cos (x)}$ by cancelling the common factors.

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Step 6.3.1.1

Cancel the common factor.

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

Step 6.3.1.2

Rewrite the expression.

$\frac{\sin (x)}{\frac{\sin (x)}{1}}$

Step 6.3.2

Divide $\sin (x)$ by $1$ .

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

Step 6.4

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

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

$1$

Step 7

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

$\frac{\sin (- x)}{\cos (- x) \tan (- x)} = 1$ is an identity