Show directly that a2 b2c2 d2 ac bd2 ad bc2 for any inte

Show directly that (a^2 + b^2)(c^2 + d^2) = (ac -bd)^2 (ad + bc)^2, for any integers a, b, c, d. This is a result we deduced in class using complex numbers

Solution

L.H.S.=(a²+b²)(c²+d²) = a²c²+a²d²+b²c²+b²d²   ..(i)

R.H.S.= (ac-bd)²+(ad-bc)² = [(ac)²+(bd)²-2(ac)(bd)]+[(ad)²+(bc)²+2(ad)(bc)] {Using (x±y)²=x²+y²±2xy}

= a²c²+b²d²-2abcd+a²d²+b²c²+2abcd=a²c²+a²d²+b²c²+b²d² ...(ii)

From (i) and (ii), we get LHS=RHS,

i.e. (a²+b²)(c²+d²) = (ac-bd)²+(ad-bc)², for all integers a,b,c&d.