Verify this is an identity? cos2x+cos2y/sinx+cosy=2cosy-2sinx

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Answer 1 · 2018-04-03T09:30:22

Please see below.

We know that

#color(red)((1)cos2theta=2cos^2theta-1,or#

#color(red)((2)cos2theta=1-2sin^2theta#

We have ,

#(cos2x+cos2y)/(sinx+cosy)=2cosy-2sinx#

#LHS=(cos2x+cos2y)/(sinx+cosy)...to Apply (1)and (2)#

#=(1-2sin^2x+2cos^2y-1)/(sinx+cosy)#

#=(2cos^2y-2sin^2x)/(sinx+cosy)#

#=(2(cos^2y-sin^2x))/(sinx+cosy)#

#=(2(cosy+sinx)(cosy-sinx))/((cosy+sinx))#

#=2(cosy-sinx)#

#=2cosy-2sinx#

#=RHS#