Trigonometry Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

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

Step 1.2

Add $1$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Move $1$ .

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

Step 2.2

Reorder $- \sin^{2} (2 x)$ and $1$ .

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

Step 2.3

Apply pythagorean identity.

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

Step 2.4

Add $\cos^{2} (2 x)$ and $\cos^{2} (2 x)$ .

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

Step 2.5

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

$0 = 0$

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:

$(- \infty, \infty)$