Simplify (cos(theta)-sin(theta))^2

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Trigonometry

${(\cos (θ) - \sin (θ))}^{2}$

Rewrite ${(\cos (θ) - \sin (θ))}^{2}$ as $(\cos (θ) - \sin (θ)) (\cos (θ) - \sin (θ))$ .

$(\cos (θ) - \sin (θ)) (\cos (θ) - \sin (θ))$

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

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Apply the distributive property.

$\cos (θ) (\cos (θ) - \sin (θ)) - \sin (θ) (\cos (θ) - \sin (θ))$

Apply the distributive property.

$\cos (θ) \cos (θ) + \cos (θ) (- \sin (θ)) - \sin (θ) (\cos (θ) - \sin (θ))$

Apply the distributive property.

$\cos (θ) \cos (θ) + \cos (θ) (- \sin (θ)) - \sin (θ) \cos (θ) - \sin (θ) (- \sin (θ))$

Simplify and combine like terms.

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Simplify each term.

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Multiply $\cos (θ) \cos (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite using the commutative property of multiplication.

$\cos^{2} (θ) - \cos (θ) \sin (θ) - \sin (θ) \cos (θ) - \sin (θ) (- \sin (θ))$

Multiply $- \sin (θ) (- \sin (θ))$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $\sin (θ)$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $\cos (θ) \sin (θ)$ from $- \cos (θ) \sin (θ)$ .

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

Move $\sin^{2} (θ)$ .

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

Apply pythagorean identity.

$1 - 2 \cos (θ) \sin (θ)$

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Name

Name	five hundred thirty million seven hundred twenty-four thousand ninety-six

Interesting facts

530724096 has 4096 divisors, whose sum is 18106260480
The reverse of 530724096 is 690427035
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 36
Digital Root 9

Name

Name	seven hundred forty-five million five hundred twenty thousand ninety-seven

Interesting facts

745520097 has 16 divisors, whose sum is 1163740032
The reverse of 745520097 is 790025547
Previous prime number is 41

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 39
Digital Root 3

Name

Name	one billion nine hundred forty-eight million four hundred ninety-four thousand nine hundred eighty

Interesting facts

1948494980 has 64 divisors, whose sum is 5383791360
The reverse of 1948494980 is 0894948491
Previous prime number is 569

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 56
Digital Root 2

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