Algebra Examples

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

Step 1

Start on the left side.

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

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

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

Step 3.1.2

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

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Step 3.1.2.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Multiply $\frac{1}{\sin (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 4

Reorder terms.

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

Step 5

Rewrite $- \cos (x) + \cos (x) \sin (x)$ as $\frac{\sin (x) - 1}{\sec (x)}$ .

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

Step 6

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

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