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