How to show that? A(x)=4cosx.cos2x.sin4x

For each x of R we put:
A(x): Sinx+Sin3x+Sin5x+Sin7x

For each x of R we put:
A(x): Sinx+Sin3x+Sin5x+Sin7x

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Feb 10, 2018

Answer:

Please see below.

Explanation:

We use here the identity relations -

#sinA+sinB=2sin((A+B)/2)cos((A-B)/2)# and

#cosA+cosB=2cos((A+B)/2)cos((A-B)/2)#

#sinx+sin3x+sin5x+sin7x#

= #sin7x+sinx+sin5x+sin3x#

= #2sin((7x+x)/2)cos((7x-x)/2)+2sin((5x+3x)/2)cos((5x-3x)/2)#

= #2sin4xcos3x+2sin4xcosx#

= #2sin4x[cos3x+cosx]#

= #2sin4x xx 2cos((3x+x)/2)cos((3x-x)/2)#

= #4sin4xcos2xcosx#

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