Trigonometry Examples

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

Step 1

Subtract $\cos^{2} (x)$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Reorder $\sin^{2} (x) \csc^{2} (x)$ and $- \sin^{2} (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Rearrange terms.

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Simplify each term.

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Step 2.7.1

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

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

Step 2.7.2

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.7.3

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

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Step 2.7.3.1

Cancel the common factor.

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

Step 2.7.3.2

Rewrite the expression.

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

Step 2.8

Subtract $\cos^{2} (x)$ from $\cos^{2} (x)$ .

$0 = 0$

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:

$(- \infty, \infty)$