Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(\sin (x) + \cos (x))}^{2}$ as $(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ .

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

Step 2.1.2

Expand $(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Apply the distributive property.

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

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $\sin (x) \sin (x)$ .

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Step 2.1.3.1.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.3.1.1.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.3.1.1.3

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

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

Step 2.1.3.1.1.4

Add $1$ and $1$ .

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

Step 2.1.3.1.2

Multiply $\cos (x) \cos (x)$ .

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Step 2.1.3.1.2.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.3.1.2.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.1.3.1.2.3

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

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

Step 2.1.3.1.2.4

Add $1$ and $1$ .

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

Step 2.1.3.2

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 2.1.3.3

Add $\cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

Simplify each term.

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Step 2.1.6.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 2.1.6.2

Reorder $\sin (x)$ and $2$ .

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

Step 2.1.6.3

Apply the sine double-angle identity.

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

Step 2.1.7

Apply the distributive property.

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

Step 2.1.8

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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Step 2.2.1

Subtract $\sin (2 x)$ from $\sin (2 x)$ .

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

Step 2.2.2

Add $\cos (2 x)$ and $0$ .

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

Step 3

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

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

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $1$ from both sides of the equation.

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

Step 4.2

Add $1$ to both sides of the equation.

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

Step 5

Simplify the left side.

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Step 5.1

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

$0 = - 1 + 1$

Step 6

Simplify the right side.

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Step 6.1

Add $- 1$ and $1$ .

$0 = 0$

Step 7

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

All real numbers

Step 8

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$