Double angle formula is can be derived using the sum of angle formulas
cos( x + y)  = cos x * cos y - sin x * sin y
what if x = y , then all y substitute x then equation becomes
cos 2x = cos^2 x - sin^2 x
and cos 2x is what you called double angle
then manipulate the equation
using trigo identities the pythagorian theorem
cos^2 x = 1 - sin^2 x substitute to the equation
 so then you come to equation
cos 2x = (1 - sin^2 x ) - sin^2 x ; combine like terms
cos 2x = 1 - 2sin^2x

then  for your problem choose what answer do you want or the best if two equation will you give to your teacher for your home work if this is  a home work haha
lets back to your equation
cos ^2 75 - sin^2 75 is equal
cos 2(75)
or 1 - 2 sin^2 (75)
or 2cos^2 (75) - 1
either of those is
equal to - sqroot(3)/2 or just - 0.8660

i hope it helps happy valentines day!!!!!