Precalculus Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 1.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.2

Combine the opposite terms in $\cos (x) + \frac{\cos (x)}{\sin (x)} - \frac{\cos (x)}{\sin (x)}$ .

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Step 1.2.1

Subtract $\frac{\cos (x)}{\sin (x)}$ from $\frac{\cos (x)}{\sin (x)}$ .

$\cos (x) + 0 = \cos (x)$

Step 1.2.2

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

$\cos (x) = \cos (x)$

Step 2

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2

Subtract $x$ from $x$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$