How do you prove the addition formula for cos?
Which level are you on?
High school or college?
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If college.
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Then,
exp (ix)=cos x+i*sin x
exp (-ix)=cos x-i*sin x
Add,
exp(ix)+exp(-ix) =2cos x
Thus,
cos x=(exp(ix)+exp(-ix))/2=cosh (ix)
Substract,
exp(ix)-exp(ix)=2i*sin x
Thus,
sin x=(exp(ix)+exp(-ix))/2i=-i*sinh (ix)
Thus when you want to show,
cos(x+y)=cos x*sin y+sin x*cos y
Just substitute those identities into the equation and show the left and right hands match