Hello!
There is a formula %7D%7D%7D%3D%7B%5Ccos%7B%7B%5Cleft(%7Ba%7D%5Cright)%7D%7D%7D%7B%5Ccos%7B%7B%5Cleft(%7Bb%7D%5Cright)%7D%7D%7D%2B%7B%5Csin%7B%7B%5Cleft(%7Ba%7D%5Cright)%7D%7D%7D%7B%5Csin%7B%7B%5Cleft(%7Bb%7D%5Cright)%7D%7D%7D. "cos(a-b)=cos(a)cos(b)+sin(a)sin(b).")
In our case, 
and 
and we know %7D%7D%7D%3D%7B%5Csin%7B%7B%5Cleft(%5Cfrac%7B%5Cpi%7D%7B%7B4%7D%7D%5Cright)%7D%7D%7D%3D%5Cfrac%7B%5Csqrt%7B%7B%7B2%7D%7D%7D%7D%7B%7B2%7D%7D. "cos(pi/4)=sin(pi/4)=sqrt(2)/2.")
So
%7D%7D%7D%3D%7B%5Ccos%7B%7B%5Cleft(%7BA%7D%5Cright)%7D%7D%7D%7B%5Ccos%7B%7B%5Cleft(%5Cfrac%7B%5Cpi%7D%7B%7B4%7D%7D%5Cright)%7D%7D%7D%2B%7B%5Csin%7B%7B%5Cleft(%7BA%7D%5Cright)%7D%7D%7D%7B%5Csin%7B%7B%5Cleft(%5Cfrac%7B%5Cpi%7D%7B%7B4%7D%7D%5Cright)%7D%7D%7D%3D%5Cfrac%7B%5Csqrt%7B%7B%7B2%7D%7D%7D%7D%7B%7B2%7D%7D%7B%5Cleft(%7B%5Ccos%7B%7B%5Cleft(%7BA%7D%5Cright)%7D%7D%7D%2B%7B%5Csin%7B%7B%5Cleft(%7BA%7D%5Cright)%7D%7D%7D%5Cright)%7D. "cos(A-pi/4) = cos(A)cos(pi/4)+sin(A)sin(pi/4)=sqrt(2)/2(cos(A)+sin(A)).")
%7D%7D%7D "cos(A)")
is given, what about %7D%7D%7D%3F "sin(A)?")
Of course %7D%2B%7B%7B%5Csin%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7BA%7D%5Cright)%7D%3D%7B1%7D%2C "cos^2(A)+sin^2(A)=1,")
so
%7D%7D%7D%3D%5Cpm%5Csqrt%7B%7B%7B1%7D-%7B%7B%5Ccos%7D%7D%5E%7B%7B2%7D%7D%7B%5Cleft(%7BA%7D%5Cright)%7D%7D%7D%3D%5Cpm%5Csqrt%7B%7B%7B1%7D-%5Cfrac%7B%7B25%7D%7D%7B%7B169%7D%7D%7D%7D%3D%5Cpm%5Csqrt%7B%7B%5Cfrac%7B%7B144%7D%7D%7B%7B169%7D%7D%7D%7D%3D%5Cpm%5Cfrac%7B%7B12%7D%7D%7B%7B13%7D%7D. "sin(A) = +-sqrt(1-cos^2(A))=+-sqrt(1-25/169) = +-sqrt(144/169) = +-12/13.")
To select "+" or "-" we have to know something additional about 
If it resides in the II quadrant, then "+", if in the III quadrant, then "-" (it cannot reside in the I or IV quadrants because its cosine is negative).
Without any additional information, we can only state that the answer is either 
or 
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