What is the exact value of #sin((7pi)/12)-sin(pi/12)#?

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Answer 1 · 2015-02-13T22:56:44

#sin( (7Pi)/12) − sin(Pi/12) = 1/sqrt(2)#

One of the standard trig. formulas states:
#sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )#

So
#sin( (7Pi)/12) − sin(Pi/12) #
# = 2 sin( ((7Pi)/12 - (pi)/12)/2 ) cos( ((7Pi)/12 + (Pi)/12)/2 )#
# = 2 sin( Pi/4 ) cos( Pi/3 )#

Since #sin(Pi/4) = 1/( sqrt(2) )#

and #cos ( (2Pi)/3) = 1/2#

#2 sin( Pi/4 ) cos( (2Pi)/3 )#
#= (2) (1/(sqrt(2))) (1/2)#
#= 1/sqrt(2)#

Therefore
#sin( (7Pi)/12) − sin(Pi/12) = 1/sqrt(2)#