We use formulas (A) - #cosA=sin(90^@-A)#,
(B) - #cos^2A-sin^2A=cos2A#
(C) - #2sinAcosA=sin2A#,
(D) - #sinA+sinB=2sin((A+B)/2)cos((A-B)/2)# and
(E) - #sinA-sinB=2cos((A+B)/2)sin((A-B)/2)#
#(cos^2 33^@-cos^2 57^@)/(sin^2 10.5^@-sin^2 34.5^@)#
= #(cos^2 33^@-sin^2 (90^@-57^@))/((sin10.5^@+sin34.5^@)(sin10.5^@-sin34.5^@))# - used A
= #(cos^2 33^@-sin^2 33^@)/(-(2sin22.5^@cos12^@)(2cos22.5^@sin12^@))# - used D & E
= #(cos66^@)/(-(2sin22.5^@cos22.5^@xx2sin12^@cos12^@)# - used B
= #-(sin(90^@-66^@))/(sin45^@sin24^@)# - used A & C
= #-sin24^@/(1/sqrt2sin24^@)#
= #-sqrt2#
