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
