hmhm maybe this :

8^[1-cos^2(x)]+8^[cos^2(x)]=10+cos2y

=>

8
----------------- + 8^[cos^2(x)]=10+cos2y
8^[cos^2(x)]

let A=8^[cos^2(x)]

now:

8
----- + A =10+cos2y
A

8+ A^2
----------- = 10+cos2y
A


im going to prove this:


8+A^2
--------- <=9
A


=> A^2 - 9A +8 <=0

A1,2=[9+_sqrt(81-32)]/2

A1,2=8,1 (*)

+++ ;;;;;;;;;;;;;;;; ++++++++
------1---------------8-------------
;;;;;; ---------------- ;;;;;;;;;;;;;

we have now that
8+A^2
---------- <=9 (**)
A


and (|cos2y|<1) => 9<=(10-cos2y)<=11 (***)

using (**) & (***)

left side is equal to right only if both sides is equal to 9

now using (*) A1,2 = 1,8

(I) A=8^[cos^2(x)]=1
cos^2(x)=0 => {{{{ cos(x)=0 }}}}
10+cos2y=9 => {{{{ cos2y=1 }}}}

(II) A=8^[cos^2(x)]=8
cos^2(x)=1
10+cos2y=9
{{{{ cos(x)=1 }}}} or {{{{ cos(x)=-1 }}}}
{{{{ cos2y =1 }}}} {{{{ cos2y =1 }}}}

finaly:

(i) x=PI/2+k*PI ; k={0,1,........}
y=k*PI ; k={0,1,........}

(ii)x=2*k*PI ; k={0,1,........}
y=k*PI ; k={0,1,........}

(iii)x=PI+2*k*PI ; k={0,1,........}
y=k*PI ; k={0,1,........}


SEE YA