Find two special angles whose sum or difference is #75^@#. (I assume you are asking about 75 degrees, not 75 radians.), the use the identity for cosine.
Find two special angles whose sum or difference is #pi/12#
The smallest special angle most of us know it #pi/6#, so you'll want a difference for this one:
#(2 pi)/ 12 - pi/12 = pi/12#,
and I know trig values for #(2 pi)/12# because that is #pi/6#but I don;t know trig functions for #pi/12#
Try something else:
#(3 pi)/ 12 - (2 pi)/12 = pi/12#,
and #(3 pi)/ 12 = pi/4# and # (2 pi)/12 = pi/6# so I know those trig values
Now use the identity:
#sin(pi/12) = sin( pi/4 - pi/6)#
#color(white)"sssssssss"# #= sin (pi/4) cos (pi/6) - cos(pi/4)sin(pi/6)#
#color(white)"sssssssss"# #= (sqrt2/2) (sqrt3/2) - (sqrt2/2)(1/2)#
#color(white)"sssssssss"# #= (sqrt6 - sqrt2)/4#
