Trigonometry Examples

$\frac{\cos (x) \tan (x) - \sin (x)}{\cot (x)} = 0$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Subtract $\sin (x)$ from $\sin (x)$ .

$0 \frac{\sin (x)}{\cos (x)}$

Step 3.4

Multiply $0$ by $\frac{\sin (x)}{\cos (x)}$ .

$0$

Step 4

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

$\frac{\cos (x) \tan (x) - \sin (x)}{\cot (x)} = 0$ is an identity