The given: #cos ((5pi)/6)*cos ((17pi)/12)#
Derivation: Use #color(red)"Sum and Difference Formulas"#
#color(blue)"cos (x+y)=cos x cos y-sin x sin y" " "#first equation
#color(blue)"cos (x-y)=cos x cos y+sin x sin y" " " "#second equation
Add first and second equations
#cos (x+y)=cos x cos y-cancel(sin x sin y)#
#cos (x-y)=cos x cos y+cancel(sin x sin y)#
th result is
#cos (x+y)+cos (x-y)=2* cos x*cos y#
divide both sides by 2 , the result is
#1/2*cos (x+y)+1/2*cos (x-y)= cos x*cos y#
Now, Let #x=(5pi)/6# and #y=(17pi)/12# then use
# cos x*cos y=1/2*cos (x+y)+1/2*cos (x-y)#
# cos ((5pi)/6)*cos ((17pi)/12)=#
#1/2*cos ((5pi)/6+(17pi)/12)+1/2*cos ((5pi)/6-(17pi)/12)#
#=1/2*cos ((27pi)/12)+1/2*cos ((-7pi)/12)#
Take note: #cos ((-7pi)/12)=cos ((+7pi)/12)#
Also #cos((27pi)/12)=cos((24pi)/12+(3pi)/12)=cos(2pi+pi/4)=cos(pi/4)#
further simplification
#1/2*cos (pi/4)+1/2*cos ((7pi)/12)#
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