## anonymous one year ago will medal for help Simplify: (sin Θ − cos Θ)^2 + (sin Θ + cos Θ)^2

1. 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 ?

2. anonymous

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

3. 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

4. anonymous

alright how would i apply that to this problem?

5. 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

6. anonymous

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

7. Nnesha

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

8. anonymous

oh okay I understand whats next?

9. 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

10. Nnesha

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

11. anonymous

a^2 - b times a?

12. Nnesha

yes right -b(a-b) =?

13. anonymous

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

14. Nnesha

no

15. 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

16. anonymous

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

17. Nnesha

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

18. Nnesha

-b times a = ?

19. anonymous

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

20. Nnesha

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

21. anonymous

-2b?

22. Nnesha

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

23. anonymous

oh right sorry it would just be 2b

24. anonymous

or b^2

25. Nnesha

yes right

26. anonymous

the b^2 is right or 2b?

27. Nnesha

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

28. Nnesha

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

29. anonymous

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

30. Nnesha

-ab-ab=??

31. anonymous

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

32. 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 $\rm x^1 \times x^1= x^{1+1}$

33. Nnesha

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

34. anonymous

alright i understand that

35. Nnesha

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

36. anonymous

yes?

37. Nnesha

i'm asking yes or no ?

38. anonymous

yes

39. Nnesha

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

40. 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!

41. anonymous

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

42. Nnesha

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

43. anonymous

I'm sorry I really don't understand?

44. Nnesha

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

45. 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

46. anonymous

would it just be a^2+b^2

47. Nnesha

no llook at the drawing ^^^^

48. anonymous

a^2+2b^2+2?

49. Nnesha

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

50. anonymous

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

51. 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

52. anonymous

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

53. Nnesha

how did you get 2b ?

54. anonymous

because of b times b

55. anonymous

but that would b^2 wouldn't it

56. Nnesha

yes right so how did you get 2b ?

57. Nnesha

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

58. anonymous

|dw:1439572494627:dw|