Trigonometry addition formula cos(a-b)=cos a cos b + sin a sin b 8080598

We are going to show that for any angles a, b the trigonometry formula cos (a-b)=cos a cos b + sin a sin b

We consider the demonstration of cos (a+b)=cos a cos b - sin a sin b as established.

It follows that:

$$\mathrm{\forall }x,y\in \mathbb{R},\cos (x+y)=\cos x\cos y-\sin x\sin y$$

In particular, by making the change of variable $x=a$, and $y=-b$

$$\cos (x+y)=\cos (a-b)=\cos a\cos (-b)-\sin a\sin (-b)$$

Since the cosine function is even:

$$\cos (-b)=\cos b$$

and the sine function is odd:

$$\sin (-b)=-\sin b$$

we have:

$$\cos (a-b)=\cos a\cos b-\sin a\times -\sin b$$

We conclude:

$$\mathrm{\forall }a,b\in \mathbb{R},\cos (a-b)=\cos a\cos b+\sin a\sin b$$