Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$(\sin (y) + \cos (y)) (\sin (y) + \cos (y)) - 1$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$\sin (y) (\sin (y) + \cos (y)) + \cos (y) (\sin (y) + \cos (y)) - 1$

Step 1.2.2

Apply the distributive property.

$\sin (y) \sin (y) + \sin (y) \cos (y) + \cos (y) (\sin (y) + \cos (y)) - 1$

Step 1.2.3

Apply the distributive property.

$\sin (y) \sin (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos (y) \cos (y) - 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $\sin (y) \sin (y)$ .

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

Multiply $\cos (y) \cos (y)$ .

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Step 1.3.1.2.1

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

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

Step 1.3.1.2.2

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

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

Step 1.3.1.2.3

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

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

Step 1.3.1.2.4

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

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

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

Step 1.5

Apply pythagorean identity.

$1 + 2 \cos (y) \sin (y) - 1$

Step 1.6

Simplify each term.

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Step 1.6.1

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

$1 + \sin (y) (2 \cos (y)) - 1$

Step 1.6.2

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

$1 + 2 \cdot \sin (y) \cos (y) - 1$

Step 1.6.3

Apply the sine double-angle identity.

$1 + \sin (2 y) - 1$

Step 2

Combine the opposite terms in $1 + \sin (2 y) - 1$ .

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Step 2.1

Subtract $1$ from $1$ .

$0 + \sin (2 y)$

Step 2.2

Add $0$ and $\sin (2 y)$ .

$\sin (2 y)$