Simplify #cos x - cos 2x + sin 2x - sin 3x = #?

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Simplify #cos x - cos 2x + sin 2x - sin 3x = #?

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Answer 1 · 2017-02-18T04:09:29

Use these trig identities:
#cos a - cos b= - 2sin ((a + b)/2)sin ((a - b)/2)#
#sin a - sin b = 2cos ((a + b)/2)sin ((a - b)/2)#
In this case:
#cos x - cos 2x = 2sin ((3x)/2)sin (x/2)# (1)
#sin 2x - sin 3x = -2cos ((5x)/2)sin (x/2)# (2)
Add (1) and (2).
Put #sin (x/2)# into common factor, we get:
#cos x - cos 2x + sin 2x - sin 3x = #
#= 2sin (x/2)[sin ((3x)/2) - cos ((5x)/2)]#

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Simplify #cos x - cos 2x + sin 2x - sin 3x = #?

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