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if a=cosx +isinx and b =cosy +isiny then find the values of cos(x+y) and cos (x-y) in terms of a and b
by eulers formula, we havea=e^ix and b=e^iynow, a*b=e^i(x+y)=cos(x+y)+isin(x+y)and 1/ab=e^-i(x+y)=cos(x+y) – isin(x+y) (because cos( – z)=cosz and sin( – z)= – sinz)adding, we get cos(x+y)= (ab+1/ab)/2now, a/b=e^i(x-y)=cos(x-y)+isin(x-y)and b/a=e^i(y-x)=cos(x-y) – isin(x-y) (because cos( – z)=cosz and sin( – z)= – sinz)adding, we havecos(x-y)= (a/b+b/a)/2
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