Trigonometry Examples

$4 \sin^{4} (x) = 1 - 2 \cos (2 \cdot x) + \cos^{2} (2 \cdot x)$

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.

$4 \sin^{4} (x) - 1 = - 2 \cos (2 \cdot x) + \cos^{2} (2 \cdot x)$

Step 1.2

Add $2 \cos (2 \cdot x)$ to both sides of the equation.

$4 \sin^{4} (x) - 1 + 2 \cos (2 \cdot x) = \cos^{2} (2 \cdot x)$

Step 1.3

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

$4 \sin^{4} (x) - 1 + 2 \cos (2 \cdot x) - \cos^{2} (2 \cdot x) = 0$

Step 2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$4 \sin^{4} (x) - 1 + 2 (1 - 2 \sin^{2} (x)) - \cos^{2} (2 \cdot x) = 0$

Step 3

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$4 \sin^{4} (x) - 1 + 2 (1 - 2 \sin^{2} (x)) - {(1 - 2 \sin^{2} (x))}^{2} = 0$

Step 4

Add $1$ to both sides of the equation.

$4 \sin^{4} (x) + 2 (1 - 2 \sin^{2} (x)) - {(1 - 2 \sin^{2} (x))}^{2} = 1$

Step 5

Simplify the left side.

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Step 5.1

Simplify $4 \sin^{4} (x) + 2 (1 - 2 \sin^{2} (x)) - {(1 - 2 \sin^{2} (x))}^{2}$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the distributive property.

$4 \sin^{4} (x) + 2 \cdot 1 + 2 (- 2 \sin^{2} (x)) - {(1 - 2 \sin^{2} (x))}^{2} = 1$

Step 5.1.1.2

Multiply $2$ by $1$ .

$4 \sin^{4} (x) + 2 + 2 (- 2 \sin^{2} (x)) - {(1 - 2 \sin^{2} (x))}^{2} = 1$

Step 5.1.1.3

Multiply $- 2$ by $2$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - {(1 - 2 \sin^{2} (x))}^{2} = 1$

Step 5.1.1.4

Rewrite ${(1 - 2 \sin^{2} (x))}^{2}$ as $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - ((1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))) = 1$

Step 5.1.1.5

Expand $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ using the FOIL Method.

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Step 5.1.1.5.1

Apply the distributive property.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 (1 - 2 \sin^{2} (x)) - 2 \sin^{2} (x) (1 - 2 \sin^{2} (x))) = 1$

Step 5.1.1.5.2

Apply the distributive property.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 \cdot 1 + 1 (- 2 \sin^{2} (x)) - 2 \sin^{2} (x) (1 - 2 \sin^{2} (x))) = 1$

Step 5.1.1.5.3

Apply the distributive property.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 \cdot 1 + 1 (- 2 \sin^{2} (x)) - 2 \sin^{2} (x) \cdot 1 - 2 \sin^{2} (x) (- 2 \sin^{2} (x))) = 1$

Step 5.1.1.6

Simplify and combine like terms.

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Step 5.1.1.6.1

Simplify each term.

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Step 5.1.1.6.1.1

Multiply $1$ by $1$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 + 1 (- 2 \sin^{2} (x)) - 2 \sin^{2} (x) \cdot 1 - 2 \sin^{2} (x) (- 2 \sin^{2} (x))) = 1$

Step 5.1.1.6.1.2

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

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) \cdot 1 - 2 \sin^{2} (x) (- 2 \sin^{2} (x))) = 1$

Step 5.1.1.6.1.3

Multiply $- 2$ by $1$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) - 2 \sin^{2} (x) (- 2 \sin^{2} (x))) = 1$

Step 5.1.1.6.1.4

Rewrite using the commutative property of multiplication.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) - 2 \cdot - 2 \sin^{2} (x) \sin^{2} (x)) = 1$

Step 5.1.1.6.1.5

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 5.1.1.6.1.5.1

Move $\sin^{2} (x)$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) - 2 \cdot - 2 (\sin^{2} (x) \sin^{2} (x))) = 1$

Step 5.1.1.6.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) - 2 \cdot - 2 {\sin (x)}^{2 + 2}) = 1$

Step 5.1.1.6.1.5.3

Add $2$ and $2$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) - 2 \cdot - 2 \sin^{4} (x)) = 1$

Step 5.1.1.6.1.6

Multiply $- 2$ by $- 2$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 2 \sin^{2} (x) - 2 \sin^{2} (x) + 4 \sin^{4} (x)) = 1$

Step 5.1.1.6.2

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

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - (1 - 4 \sin^{2} (x) + 4 \sin^{4} (x)) = 1$

Step 5.1.1.7

Apply the distributive property.

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - 1 \cdot 1 - (- 4 \sin^{2} (x)) - (4 \sin^{4} (x)) = 1$

Step 5.1.1.8

Simplify.

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Step 5.1.1.8.1

Multiply $- 1$ by $1$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - 1 - (- 4 \sin^{2} (x)) - (4 \sin^{4} (x)) = 1$

Step 5.1.1.8.2

Multiply $- 4$ by $- 1$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - 1 + 4 \sin^{2} (x) - (4 \sin^{4} (x)) = 1$

Step 5.1.1.8.3

Multiply $4$ by $- 1$ .

$4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - 1 + 4 \sin^{2} (x) - 4 \sin^{4} (x) = 1$

Step 5.1.2

Simplify by adding terms.

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Step 5.1.2.1

Combine the opposite terms in $4 \sin^{4} (x) + 2 - 4 \sin^{2} (x) - 1 + 4 \sin^{2} (x) - 4 \sin^{4} (x)$ .

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Step 5.1.2.1.1

Add $- 4 \sin^{2} (x)$ and $4 \sin^{2} (x)$ .

$4 \sin^{4} (x) + 2 + 0 - 1 - 4 \sin^{4} (x) = 1$

Step 5.1.2.1.2

Add $4 \sin^{4} (x) + 2$ and $0$ .

$4 \sin^{4} (x) + 2 - 1 - 4 \sin^{4} (x) = 1$

Step 5.1.2.1.3

Subtract $4 \sin^{4} (x)$ from $4 \sin^{4} (x)$ .

$0 + 2 - 1 = 1$

Step 5.1.2.1.4

Add $0$ and $2$ .

$2 - 1 = 1$

Step 5.1.2.2

Subtract $1$ from $2$ .

$1 = 1$

Step 6

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$