How can you prove that: (cos x + cos y)² + (sin x + sin y)² = 4*cos² (xy/2) I think you can find the curious product (a+b)² in the left-hand side to use. And in the right-hand side, I suspect you can apply Carnot’s formula for cos². But can you explain the exact solution?
Answer
You can indeed start with the left side:
If you work out the squares you get:
cos2x + cos2y + 2 cos x cos y + sin2x + sin2y + 2 sin x sin y = 2 + 2 cos(xy)
(using the fundamental formula of trigonometry, and the formula for the cosine of a difference of two angles)
On the other hand, this also applies: cos2a = ( 1 + cos 2a ) / 2
We do this for a = (xy)/2 in the right-hand side:
4 cos2 ((xy)/2) = 4 ( 1 + cos(xy) ) / 2 = 2 + cos(xy)
Both members are therefore equal to the same thing and therefore also equal to each other
Answered by
prof.dr. Paul Hellings
Department of Mathematics, Fac. IIW, KU Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/
.