I'm assuming T and ¿ are both supposed to be theta

x=rcosT

y=rsinT

(1) u = x+y so u = rcosT + rsinT

(2) v = x-y so v = rcosT - rsinT

Now work it backwards

vr = ucosT +vsinT

So

(rcosT - rsinT)*r = (rcosT + rsinT)cosT + (rcosT - rsinT)sinT

Now simplify that and remember your trig identities (cos^2(x) + sin^2(x) = 1, etc) and you should eventually get down to something like 1=1. This will prove it.

x=rcosT

y=rsinT

(1) u = x+y so u = rcosT + rsinT

(2) v = x-y so v = rcosT - rsinT

Now work it backwards

vr = ucosT +vsinT

So

(rcosT - rsinT)*r = (rcosT + rsinT)cosT + (rcosT - rsinT)sinT

Now simplify that and remember your trig identities (cos^2(x) + sin^2(x) = 1, etc) and you should eventually get down to something like 1=1. This will prove it.