Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Move $- \csc^{2} (2 x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Rearrange terms.

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

Step 2.8

Apply pythagorean identity.

$- 1 \cdot 1 + 1 = 0$

Step 2.9

Multiply $- 1$ by $1$ .

$- 1 + 1 = 0$

Step 2.10

Add $- 1$ and $1$ .

$0 = 0$

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:

$(- \infty, \infty)$