Trigonometry Examples

$\sec (x) - \sin (x) \tan (x) = \cos (x)$

Step 1

Simplify $\sec (x) - \sin (x) \tan (x)$ .

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Step 1.1

Simplify terms.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.1.3.3

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.1.3.5

Add $1$ and $1$ .

$\frac{1}{\cos (x)} - \frac{\sin^{2} (x)}{\cos (x)} = \cos (x)$

Step 1.1.2

Combine the numerators over the common denominator.

$\frac{1 - \sin^{2} (x)}{\cos (x)} = \cos (x)$

Step 1.2

Apply pythagorean identity.

$\frac{\cos^{2} (x)}{\cos (x)} = \cos (x)$

Step 1.3

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 1.3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Multiply by $1$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.3.2.4

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

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

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:

$(- \infty, \infty)$