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Simplify: (sin Θ − cos Θ)^2 + (sin Θ + cos Θ)^2

- anonymous

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Simplify: (sin Θ − cos Θ)^2 + (sin Θ + cos Θ)^2

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- schrodinger

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- Nnesha

let sin x= a
and cos = b
to make it look s easier :P \[\huge\rm (a-b)^2+(a+b)^2\]
familiar wit the foil method ?

- anonymous

ive heard of it before but I cant remember how it goes

- Nnesha

alright
(a-b)^2 is same as (a-b)(a-b)|dw:1439569755635:dw|
first multiply first term with the 2nd parentheses
and then multiply 2nd term of 1st parentheses by 2nd one

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## More answers

- anonymous

alright how would i apply that to this problem?

- Nnesha

so sin = a
b= b
(a-b)(a-b) apply the foil method \[\huge\rm \color{reD}{a}(a-b) \color{reD}{- b}(a-b)\] distribute

- anonymous

I'm sorry im still confused? are we to plug in numbers or just the letters?

- Nnesha

well we don't have to
i just change cos and sin with a b to make it a little bit easier

- anonymous

oh okay I understand whats next?

- Nnesha

(a-b)(a-b) apply the foil method \[\huge\rm \color{reD}{a}(a-b) \color{reD}{- b}(a-b)\] distribute (a-b) by a
and then (a-b) by -b

- Nnesha

|dw:1439570120307:dw|
multiply both terms by a
a times a - a times b = ?

- anonymous

a^2 - b times a?

- Nnesha

yes right
-b(a-b) =?

- anonymous

ot would it just be (a-b)^2

- Nnesha

no

- Nnesha

\(\color{blue}{\text{Originally Posted by}}\) @macim
a^2 - b times a?
\(\color{blue}{\text{End of Quote}}\)
this is right for first part

- anonymous

oh alright! and then it would be a times b -b^2?

- Nnesha

well it is -b(a-b)
signs!!!

- Nnesha

-b times a = ?

- anonymous

wouldn't it just equal -b times a because you cant multiply those variables?

- Nnesha

yes negative ab
-ab so -b times -b = ??

- anonymous

-2b?

- Nnesha

no b times b is b^2 is right but there is sign mistake
negative times negative = ?

- anonymous

oh right sorry it would just be 2b

- anonymous

or b^2

- Nnesha

yes right

- anonymous

the b^2 is right or 2b?

- Nnesha

b^2 when you multiply same bases you should add their exponents

- Nnesha

now write it together|dw:1439570550698:dw|
\[\huge\rm a^2-ab-ab+b^2\] combine like terms

- anonymous

would you just end up with a^2+b^2

- Nnesha

-ab-ab=??

- anonymous

i thought they would just cancel each other out. but if they don't wouldn't it be ab^2

- Nnesha

no
both terms are same and same sign so you can't** cancel them
\[\rm a-a= 0~~~-a-a=-2a\]
when you add or subtract like terms just simply add or subtract their coefficients
and when you multiply same bases then you should add their exponents \[\rm x^1 \times x^1= x^{1+1}\]

- Nnesha

-ab-ab is same as \[\huge\rm \color{red}{-1}ab\color{red}{-1}ab=(-1-1)ab\]

- anonymous

alright i understand that

- Nnesha

so it's equal to -2ab ? right or no?

- anonymous

yes?

- Nnesha

i'm asking yes or no ?

- anonymous

yes

- Nnesha

-ab-ab is same as \[\huge\rm \color{red}{-1}ab\color{red}{-1}ab=(-1-1)ab=-2ab\]

- Nnesha

so \[\huge\rm (a-b)^2=a^2-2ab+b^2\]
now 2nd part of the question
(a+b)^2 which is same as (a+b)(a+b) foil it!

- anonymous

foil (a-b)^2=a^2-2ab+b^2?

- Nnesha

yea that's for (a-b)^2
now foil (a+b)^2

- anonymous

I'm sorry I really don't understand?

- Nnesha

like i said (a+b)^2 is same as (a+b)(a+b) now just multiply like we did for (a-b)^2

- Nnesha

or in other words square of first number and square of 2nd number
and multiply both terms by 2
here is an example
|dw:1439571636107:dw|
just like this

- anonymous

would it just be a^2+b^2

- Nnesha

no llook at the drawing ^^^^

- anonymous

a^2+2b^2+2?

- Nnesha

|dw:1439571972434:dw|
there should be 3 terms

- anonymous

so would I have to make up another because we only have 2

- Nnesha

well you don't make up
you need to know how to foil method works
let's try box method to foil binomial |dw:1439572122692:dw|
write one parentheses at the top of the box and one left side
now start multiplying terms

- anonymous

so a^2+b^2+ab+2b

- Nnesha

how did you get 2b ?

- anonymous

because of b times b

- anonymous

but that would b^2 wouldn't it

- Nnesha

yes right
so how did you get 2b ?

- Nnesha

|dw:1439572443606:dw|
now multiply a+b by b and write it down in the box

- anonymous

|dw:1439572494627:dw|