SOLUTION: Could you please help me prove that sin(x+y)sin(x-y)=cos^2y-cos^2x? This is what I have done so far... sin(x+y)sin(x-y)can be changed to(sinxcosy+sinycosx)(sinxcosy-sinycosx

Algebra ->  Trigonometry-basics -> SOLUTION: Could you please help me prove that sin(x+y)sin(x-y)=cos^2y-cos^2x? This is what I have done so far... sin(x+y)sin(x-y)can be changed to(sinxcosy+sinycosx)(sinxcosy-sinycosx      Log On


   

Question 253255: Could you please help me prove that sin(x+y)sin(x-y)=cos^2y-cos^2x?
This is what I have done so far...
sin(x+y)sin(x-y)can be changed to(sinxcosy+sinycosx)(sinxcosy-sinycosx)
Also, cos^2y-cos^2x can be changed to 1-sin^y-1-sin^2x
I'm not sure if I am going about this the right way, so if not, could you please point me in the right direction?
Found 3 solutions by jim_thompson5910, drk, Greenfinch:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Recall that . So this means that



... Start with the given equation.


... Use the identities given above to expand the left side


... FOIL


... Distribute the exponent.


... Use the identity


... Distribute


... Combine like terms.


Since this equation is true for all values of x and y, this shows that the original equation is true for all values of x and y.


So this verifies that is an identity.

Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
We have to prove:
(i) sin(x+y)*sin(x-y))=(cos^2y-cos^2x
I will simplify the left into the right.
identity: sin (x+y) = sinxcosy + sinycosx
identity: sin(x-y) = sinxcosy - sinycosx
from (i) we get
(ii) %28sinxcosy+%2B+sinycosx%29%2A%28sinxcosy+-+sinycosx%29
multiply and we get
(iii) sin^2x*cos^2y - sin^2y*cos^2x
identity: sin^2x = 1 - cos^2x
identity: sin^2y = 1 - cos^2y
(iii) becomes
(iv) (1-cos^2x)(cos^2y) - (1-cos^2y)(cos^2x)
simplifying (iv) we get
(v) cos^2y - cos^2x
QED

Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
You have the right start.
(sinx cosy + sinycosx)(sinxcosy - sinycosx) which is difference of 2 squares so
(sin^2 x cos^2 y)- (sin^2 y cos^2 x) when multiplied out
Now sin^2 = 1 - cos^2 so, rewriting
cos^2 y ( 1 - cos^2 x) - cos^2 x (1 - cos^2 y) and expanding
cos^2 y - cos^2 y cos^2 x - cos^2 x + cos^2 x cos^2 y then adding up
cos^2 y - cos^2 x