How to prove the identity? (sin(3x))/sin(x) +(cos(3x)/cos(x))=4(1-2sin^2(x)

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Answer 1 · 2018-06-18T14:53:37

Please see the proof below.

We need

#sin(a+b)=sinacosb+sinbcosa#

#sin2x=2sinxcosx#

#cos2x=1-2sin^2x#

Therefore,

#LHS=sin(3x)/sinx+cos(3x)/cosx#

Putting on the same denominator

#=(sin3xcosx+cos3xsinx)/(sinxcosx)#

#=(sin(3x+x))/(sin(2x)/2)#

#=(2sin(4x))/sin(2x)#

#=(4cancelsin(2x)cos(2x))/(cancelsin(2x))#

#=4cos(2x)#

#=4(1-2sin^2x)#

#=RHS#

#QED#