How can we prove sin(90° - A)= cos A?

We must use the identity:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

However, there is already an A in your original 
problem, so to avoid confusion, first rewrite 
the above identity using different letters, say 
U and V. Then instead of the identity

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

after substituting U for A and V for B, we have
the identity:

sin(U - V) = sin(U)cos(V) - cos(U)sin(V)

This is necessary to do when a formula you 
want to use contains a letter that is also
contained in the expression you are wanting to 
use it in: 

So to do your problem, 

                                  sin(90° - A) = cos(A)

rewrite the left side using the identity
sin(U - V) = sin(U)cos(V) - cos(U)sin(V)
with U = 90° and V = A
                                                
               sin(90°)cos(A) - cos(90°)sin(A) = cos(A)

We use the fact that sin(90°) = 1 and cos(90°) = 0
and substitute thes values:
                                               
                    (1)cos(A) - (0)sin(A) = cos(A)
                       cos(A) -      0    = cos(A)      
                                   cos(A) = cos(A)
 
Edwin