SOLUTION: how do i verify the identity of sin(x+pi/2)=cos x

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Question 1149958: how do i verify the identity of sin(x+pi/2)=cos x
Found 3 solutions by Alan3354, Edwin McCravy, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
how do i verify the identity of sin(x+pi/2)=cos x
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Use sin(A + B)
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If you don't know that, look at "Half angle formulas" on Wikipedia

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Use the formula sin%28A%2BB%29=sin%28A%29cos%28B%29%2Bcos%28A%29sin%28B%29

on the left side:

sin%28x%2Bpi%2F2%29=cos%28x%29

sin%28x%29cos%28pi%2F2%29%2Bcos%28x%29sin%28pi%2F2%29

sin%28x%29%280%29%2Bcos%28x%29%281%29

cos%28x%29

Edwin

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
.

Apply the addition formula for sine


    sin(a+b) - sin(a)*cos(b) + cos(a)*sin(b).     (1)


It is one of the basic formula of Trigonometry; it is valid for all angles  "a" and  "b".


Put  a = x  and  b = pi%2F2.  You will get


    sin%28x+%2B+pi%2F2%29 = sin%28x%29%2Acos%28pi%2F2%29 + cos%28x%29%2Asin%28pi%2F2%29.    (2)


Take into account that   cos%28pi%2F2%29 = 0,   sin%28pi%2F2%29 = 1.

Then you can continue the line (2) in this way


         sin%28x+%2B+pi%2F2%29 = sin%28x%29%2Acos%28pi%2F2%29 + cos%28x%29%2Asin%28pi%2F2%29 = sin(x)*0 + cos(x)*1 = cos(x).


Thus the identity  sin%28x%2Bpi%2F2%29 = =cos x  is proved for all angles  x.

Solved, explained and completed.