How to prove that sin(22,5°)cos(22,5°)-cos^2(22,5°)=-(1/2) ?

1 Answer
Oct 8, 2017

See the proof below

Explanation:

Apply the following

#sin2a=2sinacosa#

#cos2a=2cos^2a-1#

#cos^2a=(1+cos2a)/2#

#sin45=cos45=sqrt2/2#

Therefore,

#LHS=sin22.5cos22.5-cos^2 22.5#

#=1/2sin(2xx22.5)-1/2(1+cos(2xx22.5))#

#=1/2sin45-1/2(1+cos45)#

#=1/2*sqrt2/2-1/2(1+sqrt2/2)#

#=sqrt2/4-1/2-sqrt2/4#

#=-1/2#

#=RHS#

#QED#