How do you find the exact value of #cos40sin55-cos55sin40# using the sum and difference, double angle or half angle formulas?

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Answer 1 · 2016-10-31T13:05:04

#(sqrt3-1)/(2sqrt2)#

Use trigonometric identity:
#cos(a+b)=cosacosb-sinasinb#, we have
#sin(55-40)=sin55cos40-sin40cos55 =sin15#

Find #sin15#, by using trigonometric identity:

#2sin^2(15)=1-cos30=1-sqrt3/2=(2-sqrt3)/2#
#sin^2(15)=(2-sqrt3)/4#
#sin(15)=+-sqrt(2-sqrt3)/2#

Only the positive value is accepted because sin15 is positive.

Finally,
#cos 40.sin 55 - cos 55.sin 40 = sin 15 = sqrt(2-sqrt3)/2 #

= #(sqrt(4-2sqrt3))/(2sqrt2)#

= #(sqrt((sqrt3)^2+1^2-2sqrt3))/(2sqrt2)#

= #(sqrt3-1)/(2sqrt2)#