Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = - \sin (x + 1)$

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

$\cos (x) (\cos (x) - \sin (x)) + \sin (x) (\cos (x) - \sin (x)) = - \sin (x + 1)$

Step 1.3.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) (\cos (x) - \sin (x)) = - \sin (x + 1)$

Step 1.3.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x)) = - \sin (x + 1)$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 1.4.1.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

$\cos (x) \cos (x) - \cos (x) \sin (x) + \cos (x) \sin (x) + \sin (x) (- \sin (x)) = - \sin (x + 1)$

Step 1.4.1.2

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

$\cos (x) \cos (x) + 0 + \sin (x) (- \sin (x)) = - \sin (x + 1)$

Step 1.4.1.3

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

$\cos (x) \cos (x) + \sin (x) (- \sin (x)) = - \sin (x + 1)$

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

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

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

Step 1.4.2.1.2

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

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

Step 1.4.2.1.3

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

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

Step 1.4.2.1.4

Add $1$ and $1$ .

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

Step 1.4.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.2.3

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

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Step 1.4.2.3.1

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

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

Step 1.4.2.3.2

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

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

Step 1.4.2.3.3

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

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

Step 1.4.2.3.4

Add $1$ and $1$ .

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

Step 1.5

Apply the cosine double-angle identity.

$\cos (2 x) = - \sin (x + 1)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 1.23746299 + 2 π n, 2.57079632 + 2 π n, 3.33185809 + 2 π n, 5.42625319 + 2 π n$ , for any integer $n$

Step 3