Question 1140881
{{{2sin^2(45-A)=1-sin(2A )}}}


Manipulating left side:


{{{2sin^2(45-A)}}}


since {{{sin(45-A)= (sqrt(2)/2)cos(A)-(sqrt(2)/2)sin(A)}}}, we have


={{{2((sqrt(2)/2)cos(A)-(sqrt(2)/2)sin(A))^2}}}...simplify, factor out {{{(1/2)^2=1/4}}}

={{{2(1/4)(sqrt(2)cos(A)-sqrt(2)sin(A))^2}}}...simplify, factor out {{{(sqrt(2))^2=2}}}


={{{cross(2)(1/cross(4)2)(2(cos(A)-sin(A))^2)}}}


={{{(1/cross(2))(cross(2)(cos(A)-sin(A))^2)}}}


={{{(cos(A)-sin(A))^2}}}.......expand


={{{cos^2(A)-2sin(A)cos(A)+sin^2(A)}}}.........use identity {{{sin^2(A))+cos^2(A)=1}}}

={{{1-2sin(A)cos(A)}}} .......use identity {{{2sin(A)cos(A) =sin(2A )}}}


={{{1-sin(2A )}}}