Precalculus Examples

$\frac{1 + \csc (x)}{\sec (x)} - \cot (x) = \cos (x)$

Step 1

Start on the left side.

$\frac{1 + \csc (x)}{\sec (x)} - \cot (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Multiply $\cos (x)$ by $1$ .

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

Step 3.1.4

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

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

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

Step 3.4

Divide $0$ by $\sin (x)$ .

$\cos (x) + 0$

Step 3.5

Add $\cos (x)$ and $0$ .

$\cos (x)$

Step 4

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

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