Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{\cos (x) (\tan (x) - \sec (x))}{1 - \csc (x)}$

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Write $1$ as a fraction with a common denominator.

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.3

Combine $\cos (x)$ and $\frac{\sin (x) - 1}{\cos (x)}$ .

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

Step 3.4

Reduce the expression by cancelling the common factors.

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Step 3.4.1

Reduce the expression $\frac{\cos (x) (\sin (x) - 1)}{\cos (x)}$ by cancelling the common factors.

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Step 3.4.1.1

Cancel the common factor.

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

Step 3.4.1.2

Rewrite the expression.

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

Step 3.4.2

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

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

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.6

Cancel the common factor of $\sin (x) - 1$ .

$\sin (x)$

Step 4

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

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