Trigonometry Examples

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

Step 1

Subtract $\sin (x) \cos (x)$ from both sides of the equation.

$(1 - \cos^{2} (x)) \cot (x) - \sin (x) \cos (x) = 0$

Step 2

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

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Step 2.1

Factor $1$ out of $(1 - \cos^{2} (x)) \cot (x) - \sin (x) \cos (x)$ .

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Step 2.1.1

Factor $1$ out of $(1 - \cos^{2} (x)) \cot (x)$ .

$1 ((1 - \cos^{2} (x)) \cot (x)) - \sin (x) \cos (x) = 0$

Step 2.1.2

Factor $1$ out of $- \sin (x) \cos (x)$ .

$1 ((1 - \cos^{2} (x)) \cot (x)) + 1 (- \sin (x) \cos (x)) = 0$

Step 2.1.3

Factor $1$ out of $1 ((1 - \cos^{2} (x)) \cot (x)) + 1 (- \sin (x) \cos (x))$ .

$1 ((1 - \cos^{2} (x)) \cot (x) - \sin (x) \cos (x)) = 0$

Step 2.2

Apply pythagorean identity.

$1 (\sin^{2} (x) \cot (x) - \sin (x) \cos (x)) = 0$

Step 2.3

Multiply $\sin^{2} (x) \cot (x) - \sin (x) \cos (x)$ by $1$ .

$\sin^{2} (x) \cot (x) - \sin (x) \cos (x) = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

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

$\sin^{2} (x) \frac{\cos (x)}{\sin (x)} - \sin (x) \cos (x) = 0$

Step 2.4.2

Cancel the common factor of $\sin (x)$ .

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Step 2.4.2.1

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

$\sin (x) \sin (x) \frac{\cos (x)}{\sin (x)} - \sin (x) \cos (x) = 0$

Step 2.4.2.2

Cancel the common factor.

$\sin (x) \sin (x) \frac{\cos (x)}{\sin (x)} - \sin (x) \cos (x) = 0$

Step 2.4.2.3

Rewrite the expression.

$\sin (x) \cos (x) - \sin (x) \cos (x) = 0$

Step 2.5

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

$0 = 0$

Step 3

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$