How do you prove $cos (x + y) cos y + sin (x + y) sin y = cos x$ $cos(x+y)cosy + sin(x+y)siny = cosx$ ?

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Answer 1 · 2016-04-28T03:36:54

Please see below

Recall the trigonometrical identity

$cos (A - B) = cos A cos B + sin A sin B$ $cos(A-B)=cosAcosB+sinAsinB$

Putting $A = x + y$ $A=x+y$ and $B = y$ $B=y$ , we get

$cos (x + y - y) = cos (x + y) cos y + sin (x + y) sin y$ $cos(x+y-y)=cos(x+y)cosy+sin(x+y)siny$

or transposing LHS to RHS and vice-versa

$cos (x + y) cos y + sin (x + y) sin y = cos x$ $cos(x+y)cosy+sin(x+y)siny=cosx$

How do you prove cos(x+y)cosy + sin(x+y)siny = cosxcos(x+y)cosy+sin(x+y)siny=cosx cos(x+y)cosy + sin(x+y)siny = cosx?