How do you use the product to sum formulas to write #cos(theta-pi)sin(theta+pi)# as a sum or difference?

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Answer 1 · 2017-01-06T07:59:31

#cos(theta-pi)sin(theta+pi)=1/2sin2theta-0=sinthetacostheta#

We can use here the product formula

#cosAsinB=1/2[sin(A+B)-sin(A-B)]#

Hence #cos(theta-pi)sin(theta+pi)#

= #1/2[sin((theta-pi)+(theta+pi))-sin((theta-pi)-(theta+pi))]#

= #1/2[sin(theta-pi+theta+pi)-sin(theta-pi-theta-pi)]#

= #1/2sin2theta-sin(-2pi)]# #-# #color(red)(but)# #color(red)(as)# #sin(-2pi)=0#

= #1/2sin2theta-0#

Here, we have written as a difference, but latter component is zero

and can be simplified further as

#1/2sin2theta=1/2xx2sinthetacostheta=sinthetacostheta#