Trigonometry Examples

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

Step 1

Double negation

$\sin (x - (- 1)) \sin (x - 1)$

Step 2

Apply the difference of angles identity.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Evaluate $\cos (- 1)$ .

$(\sin (x) \cdot 0.5403023 - \cos (x) \sin (- 1)) \sin (x - 1)$

Step 3.1.2

Move $0.5403023$ to the left of $\sin (x)$ .

$(0.5403023 \cdot \sin (x) - \cos (x) \sin (- 1)) \sin (x - 1)$

Step 3.1.3

Evaluate $\sin (- 1)$ .

$(0.5403023 \sin (x) - \cos (x) \cdot - 0.84147098) \sin (x - 1)$

Step 3.1.4

Multiply $- 0.84147098$ by $- 1$ .

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) \sin (x - 1)$

Step 3.2

Separate negation.

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) \sin (x - (1))$

Step 4

Apply the difference of angles identity.

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) (\sin (x) \cos (1) - \cos (x) \sin (1))$

Step 5

Simplify each term.

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Step 5.1

Evaluate $\cos (1)$ .

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) (\sin (x) \cdot 0.5403023 - \cos (x) \sin (1))$

Step 5.2

Move $0.5403023$ to the left of $\sin (x)$ .

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) (0.5403023 \cdot \sin (x) - \cos (x) \sin (1))$

Step 5.3

Evaluate $\sin (1)$ .

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) (0.5403023 \sin (x) - \cos (x) \cdot 0.84147098)$

Step 5.4

Multiply $0.84147098$ by $- 1$ .

$(0.5403023 \sin (x) + 0.84147098 \cos (x)) (0.5403023 \sin (x) - 0.84147098 \cos (x))$

Step 6

Expand $(0.5403023 \sin (x) + 0.84147098 \cos (x)) (0.5403023 \sin (x) - 0.84147098 \cos (x))$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$0.5403023 \sin (x) (0.5403023 \sin (x) - 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x) - 0.84147098 \cos (x))$

Step 6.2

Apply the distributive property.

$0.5403023 \sin (x) (0.5403023 \sin (x)) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x) - 0.84147098 \cos (x))$

Step 6.3

Apply the distributive property.

$0.5403023 \sin (x) (0.5403023 \sin (x)) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $0.5403023 \sin (x) (0.5403023 \sin (x))$ .

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Step 7.1.1.1

Multiply $0.5403023$ by $0.5403023$ .

$0.29192658 \sin (x) \sin (x) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.1.2

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

$0.29192658 (\sin^{1} (x) \sin (x)) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.1.3

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

$0.29192658 (\sin^{1} (x) \sin^{1} (x)) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.1.4

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

$0.29192658 {\sin (x)}^{1 + 1} + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.1.5

Add $1$ and $1$ .

$0.29192658 \sin^{2} (x) + 0.5403023 \sin (x) (- 0.84147098 \cos (x)) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.2

Multiply $- 0.84147098$ by $0.5403023$ .

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.84147098 \cos (x) (0.5403023 \sin (x)) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.3

Multiply $0.5403023$ by $0.84147098$ .

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) + 0.84147098 \cos (x) (- 0.84147098 \cos (x))$

Step 7.1.4

Multiply $0.84147098 \cos (x) (- 0.84147098 \cos (x))$ .

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Step 7.1.4.1

Multiply $- 0.84147098$ by $0.84147098$ .

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 \cos (x) \cos (x)$

Step 7.1.4.2

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

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 (\cos^{1} (x) \cos (x))$

Step 7.1.4.3

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

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 (\cos^{1} (x) \cos^{1} (x))$

Step 7.1.4.4

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

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 {\cos (x)}^{1 + 1}$

Step 7.1.4.5

Add $1$ and $1$ .

$0.29192658 \sin^{2} (x) - 0.45464871 \sin (x) \cos (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 \cos^{2} (x)$

Step 7.2

Reorder the factors of $- 0.45464871 \sin (x) \cos (x)$ .

$0.29192658 \sin^{2} (x) - 0.45464871 \cos (x) \sin (x) + 0.45464871 \cos (x) \sin (x) - 0.70807341 \cos^{2} (x)$

Step 7.3

Add $- 0.45464871 \cos (x) \sin (x)$ and $0.45464871 \cos (x) \sin (x)$ .

$0.29192658 \sin^{2} (x) + 0 \cos (x) \sin (x) - 0.70807341 \cos^{2} (x)$

Step 8

Simplify each term.

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Step 8.1

Multiply $0$ by $\cos (x)$ .

$0.29192658 \sin^{2} (x) + 0 \sin (x) - 0.70807341 \cos^{2} (x)$

Step 8.2

Multiply $0$ by $\sin (x)$ .

$0.29192658 \sin^{2} (x) + 0 - 0.70807341 \cos^{2} (x)$

Step 9

Add $0.29192658 \sin^{2} (x)$ and $0$ .

$0.29192658 \sin^{2} (x) - 0.70807341 \cos^{2} (x)$