SOLUTION: Please prove this identity: sin[(pi/2 - x) + y] = cos(x-y)

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Question 1011267: Please prove this identity:
sin[(pi/2 - x) + y] = cos(x-y)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

sin%28%28pi%2F2+-+x%29+%2B+y%29+=+cos%28x-y%29
start with left side and arrive to right side:

sin%28%28pi%2F2+-+x%29%29+sin%28y%29%2Bcos%28%28pi%2F2+-+x%29%29+cos%28y%29

since
sin%28pi%2F2-x%29+=+cos%28x%29+sin%28pi%2F2%29-cos%28pi%2F2%29+sin%28x%29
cos%28pi%2F2-x%29+=+cos%28pi%2F2%29+cos%28-x%29-sin%28pi%2F2%29+sin%28-x%29

we have

=
= ...since sin%28pi%2F2%29=1 and cos%28pi%2F2%29=0, we have

=

=cos%28x%29+sin%28y%29+-cos%28y%29+sin%28-x%29+=cos%28x-y%29...since sin%28-x%29+=-sin%28x%29, we have

=cos%28x%29sin%28y%29-%28-sin%28x%29+cos%28y%29%29
=cos%28x%29sin%28y%29-sin%28x%29+cos%28y%29
=cos%28x-y%29