## chaotic_butterflies one year ago (sin Θ − cos Θ)^2 + (sin Θ + cos Θ)^2 I've worked this out multiple times, and my answer isn't a possible choice...

1. chaotic_butterflies

Answer choices: A) 1 B) 2 C) sin^2 theta D) cos^2 theta

2. Nnesha

3. Nnesha

remember these two formulas $\huge\rm (a+b)^2 = a^2 +2ab+b^2$ $\huge\rm (a-b)^2 = a^2 -2ab+b^2$

4. chaotic_butterflies

Oh goodness, I came at this a couple of different ways. At first I tried to replace sin and cos theta with regular numbers to see if the setup of the problem was some sort of rule, and that didn't come out right.

5. Nnesha

alright so let sin theta = a cos theta = b so $\huge\rm (a-b)^2 =???$

6. chaotic_butterflies

a^2 - b^2?

7. Nnesha

nope...

8. Nnesha

$$\color{blue}{\text{Originally Posted by}}$$ @Nnesha remember these two formulas $\huge\rm (a+b)^2 = a^2 +2ab+b^2$ $\huge\rm (a-b)^2 = a^2 -2ab+b^2$ $$\color{blue}{\text{End of Quote}}$$

9. Nnesha

square of first number and square of 2nd term thne MULTIPLY BOTH TERMS by 2

10. Nnesha

or in other words (a-b)^2 is same as (a-b)(a-b) so you can foil :-)

11. chaotic_butterflies

Okay, I can at least do that...

12. Nnesha

ohh nice :-)

13. Nnesha

let me know what you get :-)

14. chaotic_butterflies

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

15. chaotic_butterflies

Well, that's at least the first step... am I wrong or doesn't that look like it should cancel out? @Nnesha

16. Nnesha

yes that's right now apply foil method :-)

17. Nnesha

no no you can't cancel anything yet

18. Nnesha

let sin = x cos = y it will be easy like simple algebra :-) (x-y)(x-y ) :-)

19. chaotic_butterflies

Oh phoey... well I'll continue to solve. If you don't mind, I like to simplify by grouping. (x-y)(x-y)+(x+y)(x+y) x^2 - xy (-y) (x-y) + (x+y)(x+y) x^2 - xy - yx + y^2 + (x+ y) (x+y) x^2 - xy - yx + y^2 + x^2 + xy (y)(x+y) x^2 - xy - yx + y^2 + x^2 + xy + yx + y^2 x^4 + y^2

20. chaotic_butterflies

*y^4

21. Nnesha

alright good job thanks! now x^2 +y^2 +x^2 +y^2 here COMBINE LIKE terms when you MULTIPLY same bases THEN you should add their exponent

22. Nnesha

$\huge\rm x^2+y^2+x^2+y^2= ??$ combine like terms

23. chaotic_butterflies

Did I not already do that?

24. Nnesha

nope last step is wrong don't add their exponent here is a different COMBINE like terms : x+x = 2x Multiply x times x= x^2 there is a plus sign so you can't add their exponents

25. chaotic_butterflies

Goodness... I need to fresh up on my basic algebra >.<

26. Nnesha

alright so combine like terms meaning add coefficient of same bases

27. chaotic_butterflies

So 2x^4 + 2y^4..?

28. Nnesha

yeah but exponent should the same remember exponent rule $\large\rm x \times x = x^2$ when u MULTIPLY same bases then u should add exponents

29. chaotic_butterflies

Oh wait it would just be 2x^2 + 2y^2 wouldn't it?

30. chaotic_butterflies

31. Nnesha

yes that's right

32. chaotic_butterflies

So that leaves me with $\sin \theta ^{2} + \cos \theta ^{2}$

33. Nnesha

now 2 is common factor s take it out $\huge\rm 2x^2+2y^2$$\large\rm 2(x^2+y^2)$

34. Nnesha

yes right change back to sin and cos so remember special identity sin^2 x + cos^2 x = what ?

35. chaotic_butterflies

36. chaotic_butterflies

I honestly don't remember.

37. Nnesha

hmm that's the only you shouldn't forget $\huge\rm sin^2 \theta + \cos^2 \theta =1$

38. Nnesha

so replace sin^2 + cos ^2 by 1 :-)

39. chaotic_butterflies

So it's just 2*1 = 2?

40. Nnesha

yes right :-) 2!!

41. chaotic_butterflies

Yay! Thank you for being so detailed, it really helped

42. Nnesha

my pleasure :-) gO_od job!! :=)