Precalculus Examples

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

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the equation.

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

Step 1.2

Add $2 \sin^{2} (y)$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Move $- 1$ .

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

Step 2.2

Reorder $\cos^{2} (y)$ and $- 1$ .

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

Step 2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Rewrite $- 1 (1 - \cos^{2} (y))$ as $- (1 - \cos^{2} (y))$ .

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

Step 2.7

Apply pythagorean identity.

$- \sin^{2} (y) - \sin^{2} (y) + 2 \sin^{2} (y) = 0$

Step 2.8

Subtract $\sin^{2} (y)$ from $- \sin^{2} (y)$ .

$- 2 \sin^{2} (y) + 2 \sin^{2} (y) = 0$

Step 2.9

Add $- 2 \sin^{2} (y)$ and $2 \sin^{2} (y)$ .

$0 = 0$

Step 3

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$