Question

If Ф(x) = (a^x+a^-x) / 2 and ϴ(x) = (a^x-a^-x)/2 Prove that a)Ф(x+y)=Ф(x).Ф(y) + ϴ(x).ϴ(y) and b) ϴ(x+y)=Ф(x).ϴ(y) + Ф(y).ϴ(x)

Answer 1

a) for the first one I would play with the right hand side to show the left hand side

Answer 2

Same thing for b

Answer 3

Have you tried plugging in and doing the operations to see if you get the right hand side?

Answer 4

\[
\phi(x) = \frac 12(a^x + a^{-x}) \\
\phi(x+y) = \frac 12 (a^{x+y} + a^{-{(x+y)} }) = \frac 12(a^x*a^y + a^{-x}*a^{-y})
\]Compute the right hand side separately and show they are equal.

Answer 5

\[\frac{a^x+a^{-x}}{2} \cdot \frac{a^y+a^{-y}}{2}+\frac{a^x-a^{-x}}{2}\cdot \frac{a^y-a^{-y}}{2}\]

Answer 6

try to multiply and add like terms if they should result

Answer 7

\[\frac{a^x+a^{-x}}{2} \cdot \frac{a^y+a^{-y}}{2}+\frac{a^x-a^{-x}}{2}\cdot \frac{a^y-a^{-y}}{2} \\ \frac{a^xa^y+a^xa^{-y}+a^{-x}a^y+a^{-x}a^{-y}}{4}+\frac{a^xa^y-a^xa^{-y}-a^{-x}a^{y}+a^{-x}a^{-y}}{4} \\\]
combine like terms on top

Answer 8

from both fractions you should see a lot of cancellation

Answer 9

And you will see double of two terms

Answer 10

but guess what 2/4=1/2

Answer 11

I think I lost myself =(

Answer 12

\[\frac{a^x+a^{-x}}{2} \cdot \frac{a^y+a^{-y}}{2}+\frac{a^x-a^{-x}}{2}\cdot \frac{a^y-a^{-y}}{2} \\ \frac{a^xa^y+a^xa^{-y}+a^{-x}a^y+a^{-x}a^{-y}}{4}+\frac{a^xa^y-a^xa^{-y}-a^{-x}a^{y}+a^{-x}a^{-y}}{4} \\ \frac{a^xa^y+\cancel{a^xa^{-y}}+\cancel{a^{-x}a^y}+a^{-x}a^{-y}}{4}+\frac{a^xa^y-\cancel{a^xa^{-y}}-\cancel{a^{-x}a^{y}}+a^{-x}a^{-y}}{4} \\ \]

Answer 13

you have a^xa^y+a^xa^y which is 2a^xa^y right?

Answer 14

and a^(-x)a^(-y)+a^(-x)a^(-y)=2a^(-x)a^(-y)

Answer 15

\[\frac{a^x+a^{-x}}{2} \cdot \frac{a^y+a^{-y}}{2}+\frac{a^x-a^{-x}}{2}\cdot \frac{a^y-a^{-y}}{2} \\ \frac{a^xa^y+a^xa^{-y}+a^{-x}a^y+a^{-x}a^{-y}}{4}+\frac{a^xa^y-a^xa^{-y}-a^{-x}a^{y}+a^{-x}a^{-y}}{4} \\ \frac{a^xa^y+\cancel{a^xa^{-y}}+\cancel{a^{-x}a^y}+a^{-x}a^{-y}}{4}+\frac{a^xa^y-\cancel{a^xa^{-y}}-\cancel{a^{-x}a^{y}}+a^{-x}a^{-y}}{4} \\ \frac{2a^xa^y+2a^{-x}a^{-y}}{4} \\ =2 \frac{a^{x+y}+a^{-x-y}}{4}\]

Answer 16

reduce the 2/4

Answer 17

I was confused because your last answer already contain a step that would be enough to say that Ф (x + y) = Ф (x) .Ф (y) + Θ (x) .Θ (y) if the step it would be:
(a^x+a^-x) /2 . (a^y + a^-y) / 2 + (a^x-a^-x)/2 + (a^y - a^-y) /2

Answer 18

i'm not sure what step you are talking about

Answer 19

first step in answer above of "reduce the 2/4"