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

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Answer 1 · 2017-10-08T08:57:20

See the proof below

Apply the following

$sin2a=2sinacosa$

$cos2a=2cos^2a-1$

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

$sin45=cos45=sqrt2/2$

Therefore,

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

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

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

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

$=sqrt2/4-1/2-sqrt2/4$

$=-1/2$

$=RHS$

$QED$