Prove cos(a+b)cos(a-b)=cos^2b-sin^2a ?

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Answer 1 · 2017-04-22T09:30:17

See proof below

We need

#(x+y)(x-y)=x^2-y^2#

#cos(a+b)=cosacosb-sina sinb#

#cos(a-b)=cosacosb+sina sinb#

#cos^2a+sin^2a=1#

#cos^2b+sin^2b=1#

Therefore,

#LHS=cos(a+b)cos(a-b)#

#=(cosacosb-sina sinb)(cosacosb+sina sinb)#

#=cos^2acos^2b-sin^2a sin^2b#

#=cos^2b(1-sin^2a)-sin^2a(1-cos^2b)#

#=cos^2b-cancel(cos^2bsin^2a)-sin^2a+cancel(cos^2bsin^2a)#

#=cos^2b-sin^2a#

#=RHS#

#QED#