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

2 Answers
Mar 17, 2016

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)#

Mar 18, 2016

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)#