Basic Math Examples

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

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite

{(\sin (y) + \cos (y))}^{2}

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

(\sin (y) + \cos (y)) (\sin (y) + \cos (y))

$(\sin (y) + \cos (y)) (\sin (y) + \cos (y))$ .

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

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

Step 1.2

Expand

(\sin (y) + \cos (y)) (\sin (y) + \cos (y))

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

Tap for more steps...

Step 1.2.1

Apply the distributive property.

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

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

Step 1.2.2

Apply the distributive property.

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

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

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

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

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

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

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply

\sin (y) \sin (y)

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

Tap for more steps...

Step 1.3.1.1.1

Raise

\sin (y)

$\sin (y)$ to the power of

1

$1$ .

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

$\sin (y)$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.3.1.1.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

Tap for more steps...

Step 1.3.1.2.1

Raise

\cos (y)

$\cos (y)$ to the power of

1

$1$ .

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

$\cos (y)$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.3.1.2.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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

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

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

Step 1.3.3

Add

\cos (y) \sin (y)

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

\cos (y) \sin (y)

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

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

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

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

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

Step 1.4

Move

\cos^{2} (y)

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

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

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

Step 1.5

Apply pythagorean identity.

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

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

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Reorder

2 \cos (y)

$2 \cos (y)$ and

\sin (y)

$\sin (y)$ .

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

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

Step 1.6.2

Reorder

\sin (y)

$\sin (y)$ and

2

$2$ .

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

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

Step 1.6.3

Apply the sine double-angle identity.

1 + \sin (2 y) - 1

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

1 + \sin (2 y) - 1

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

1 + \sin (2 y) - 1

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

Step 2

Combine the opposite terms in

1 + \sin (2 y) - 1

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

Tap for more steps...

Step 2.1

Subtract

1

$1$ from

1

$1$ .

0 + \sin (2 y)

$0 + \sin (2 y)$

Step 2.2

Add

0

$0$ and

\sin (2 y)

$\sin (2 y)$ .

\sin (2 y)

$\sin (2 y)$

\sin (2 y)

$\sin (2 y)$