Trigonometry Examples

$\frac{\cot (- x)}{\csc (x)} + \cos (x) = 0$

Step 1

Start on the left side.

$\frac{\cot (- x)}{\csc (x)} + \cos (x)$

Step 2

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

$\frac{- \cot (x)}{\csc (x)} + \cos (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

Apply the reciprocal identity to $\csc (x)$ .

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

$- \cos (x) + \cos (x)$

Step 4.2

Add $- \cos (x)$ and $\cos (x)$ .

$0$

Step 5

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

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