Question

Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β

Answer 1

@ganeshie8 @pooja195 @hartnn @triciaal

Answer 2

do you know the identity cos(x-y) = ??

Answer 3

No

Answer 4

hmm alright use this \[\huge\rm cos(\color{Red}{x}-\color{blue}{y})=cos\color{Red}{x} \cos \color{blue}{y} + \sin \color{Red}{x} \sin\color{blue}{ y}\]
and
\[\huge\rm cos(\color{Red}{x}+\color{blue}{y})=cos\color{Red}{x} \cos \color{blue}{y} - \sin \color{Red}{x} \sin\color{blue}{ y}\]

Answer 5

okay but how would I solve it?

Answer 6

replace cos(a-b) and cos (a+b) with those identities

Answer 7

like for cos(a-b)\[\huge\rm \color{ReD}{cos\color{Red}{\alpha } \cos \color{blue}{\beta} + \sin \color{Red}{\alpha } \sin\color{blue}{ \beta } }- cos (\alpha + \beta)\]
use first identity for cos (a +b)

Answer 8

okay but still how would I solve it

Answer 9

replace cos (a+b) with this identity and then distribute it by -1 \[\huge\rm cos(\color{Red}{x}+\color{blue}{y})=cos\color{Red}{x} \cos \color{blue}{y} - \sin \color{Red}{x} \sin\color{blue}{ y}\]

Answer 10

cos (a+b) = cos a cos b - sin a sin b
so can substitute `cos(a+b)` for { cos a cos - sin a sin b}

Answer 11

\[\large\rm \color{ReD}{cos\color{Red}{\alpha } \cos \color{blue}{\beta} + \sin \color{Red}{\alpha } \sin\color{blue}{ \beta } }- (cos \alpha cos \beta - sin \alpha sin \beta)\]
distribute parentheses by negative one you will get the answer

Answer 12

This makes no sense to me

Answer 13

\[\large\rm \color{ReD}{cos\color{Red}{(\alpha) } \cos \color{blue}{(\beta)} + \sin \color{Red}{(\alpha) } \sin\color{blue}{ (\beta) } }- [cos (\alpha) cos (\beta) - sin( \alpha) sin (\beta)]\]

Answer 14

It's an identity, there's a proof for it, but you just have to remember that
\[\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)\]\[\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)\]

Answer 15

just like how we have algebriac identities
\[(a+b)^2=a^2+b^2+2ab\]
we have numerous trigonometric identities, you don't need to ground proving them every time, they r something you have to learn

Answer 16

go around not ground

Answer 17

it's just like algebra plugin the identities
let's say cos a = x
and cos b = y
sin a = w
sin b =z
so \[\huge\rm x y+ wz - (xy - wz ) = 2 wz\] can you prove this ?

Answer 18

@Nishant_Garg Ok I get it, so can you explain the step by step process on proving the identity of cos(a-b) - cos(a+b) = 2 sin a sin b?

Answer 19

http://www.purplemath.com/modules/idents.htm here ar all of the identities
you should write theses down n ur notes

Answer 20

these*

Answer 21

Sorry, I haven't gone through the proofs myself, in our time we were just told to learn these identities even though the proof was in the book, it was just there for satisfaction of the mind, they never ask u the proofs

Answer 22

@Nnesha I get what you saying but can you just please explain the proof process? You telling me isn't helping cause I don't know what I'm doing.. I need someone to SHOW me..

Answer 23

do you mean show how cos(a+b) = cos (a)cos(b) - sin (a) sin(b) ???? this identity proof ?

Answer 24

Weren't you taught about the proofs, if you are being given questions like these, then they must've also taught u the proofs(talking about your school), if not they must've at least taught u about these identities

Answer 25

Like work it out step by step & just tell me how you got the answer so I can work out my other problems like this on my own.

Answer 26

No this one.... cos(a-b) - cos(a+b) = 2 sin a sin b

Answer 27

\[\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)\]\[\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)\]
Using these 2 identities we get
\[\cos(a-b)-\cos(a+b)=(\cos(a)\cos(b)+\sin(a)\sin(b))-(\cos(a)\cos(b)-\sin(a)\sin(b))\]

Answer 28

i understand but i would like u to work with me
i just .... don't feel comfortable doing all work...
even if the question is simple like 2+2 = ?
anywaz do you know how to distribute ?? :=)

Answer 29

after that its just cancelling some terms adding some terms and u can get the result

Answer 30

\(\color{blue}{\text{Originally Posted by}}\) @Nnesha
\[\large\rm \color{ReD}{cos\color{Red}{(\alpha) } \cos \color{blue}{(\beta)} + \sin \color{Red}{(\alpha) } \sin\color{blue}{ (\beta) } }- [cos (\alpha) cos (\beta) - sin( \alpha) sin (\beta)]\]
\(\color{blue}{\text{End of Quote}}\)
did you understand this step so we can do next stpe ?

Answer 31

No just don't worry about it, thanks for trying!

Answer 32

i'll give it a last try :=)
let's say we have two equations