Question

$find value of cos20 + 2{sin}^{2} 55 - \sqrt{2} sin 65$

Answer 1

consider the following...
110/2=55
45+65=110

cos(A + B) = cos A cos B  −  sin A sin B

let A=B : cos 2A=1-2sin² A

let A=55  : 2sin² 55=1-cos 110

cos20-cos110 

=cos(65-45)-cos(65+45)

=cos65cos45+sin65sin45-cos65cos45+sin65sin45

=2sin65/√2

=√2sin65

using these, we can 

cos20-cos110 -√2sin65=0

cos20+1-cos110 -√2sin65=1

cos20+2sin² 55 -√2sin65=1