Trigonometry Examples

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

Step 1

Simplify $\sqrt{1 - \cos^{2} (x)}$ .

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Step 1.1

Apply pythagorean identity.

$\sin (x) = \sqrt{\sin^{2} (x)}$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

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

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:

$(- \infty, \infty)$