Question

How do you prove `sin (π/2 – x) = sin (π/2 + x)`?

Answer 1

What proofs are available depends on what facts we already have as given. One simple proof relies on the following:

• `sin(x) = -sin(x-pi)`
  -`sin(-x) = -sin(x)`

With those, we have

`sin(pi/2-x) = -sin((pi/2-x)-pi)`

`=-sin(-x-pi/2)`

`=-sin(-(pi/2+x))`

`=-(-sin(pi/2+x))`

`=sin(pi/2+x)`

Answer 2

Apply the sum identities.
`sin (pi/2 - x) = sin (pi/2).cos x - cos (pi/2).sin x = cos x`
because `sin (pi/2) = 1` and `cos (pi/2) = 0`
`sin (pi/2 + x) = sin (pi/2).cos x + cos (pi/2) sin x = cos x`
Therefor,
`sin (pi/2 - x) = cos x = sin (pi/2 + x)`