Precalculus Examples

sin (x) \cdot sec (x) = tan (x)

$\sin (x) \cdot \sec (x) = \tan (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify

sin (x) \cdot sec (x)

$\sin (x) \cdot \sec (x)$ .

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Step 1.1.1

Rewrite

sec (x)

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

sin (x) \cdot \frac{1}{cos (x)} = tan (x)

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

Step 1.1.2

Combine

sin (x)

$\sin (x)$ and

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ .

\frac{sin (x)}{cos (x)} = tan (x)

$\frac{\sin (x)}{\cos (x)} = \tan (x)$

\frac{sin (x)}{cos (x)} = tan (x)

$\frac{\sin (x)}{\cos (x)} = \tan (x)$

\frac{sin (x)}{cos (x)} = tan (x)

$\frac{\sin (x)}{\cos (x)} = \tan (x)$

Step 2

Simplify the right side.

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Step 2.1

Rewrite

tan (x)

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

\frac{sin (x)}{cos (x)} = \frac{sin (x)}{cos (x)}

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

\frac{sin (x)}{cos (x)} = \frac{sin (x)}{cos (x)}

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

Step 3

Multiply both sides of the equation by

cos (x)

$\cos (x)$ .

cos (x) \frac{sin (x)}{cos (x)} = cos (x) \frac{sin (x)}{cos (x)}

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

Step 4

Cancel the common factor of

cos (x)

$\cos (x)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Cancel the common factor of

$\cos (x)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\sin (x) = \sin (x)$

Step 6

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

$x = x$

Step 7

Move all terms containing

$x$ to the left side of the equation.

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Step 7.1

Subtract

$x$ from both sides of the equation.

$x - x = 0$

Step 7.2

Subtract

$x$ from

$x$ .

$0 = 0$

Step 8

Since

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

$x$ .

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$