Question

Prove, using vectors, that $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta .$

Answer 1

Let $\hat{a}=cos\alpha \hat{i}+sin\alpha \hat{j}\&\hat{b}=cos\beta \hat{i}-sin\beta \hat{j}$

i.e $\hat{a}$ makes angle $\alpha$ with $x$ axis & $\hat{b}$ make $-\beta$ with $x$ axis.

$\therefore$ $\measuredangle$ b/w $\hat{a}\&\hat{b}=\alpha -(-\beta )=\alpha +\beta$

Now, $\left|\hat{a}\right|=1\&\left|\hat{b}\right|=1$ [ $\because$ they are unit vectors]

Now, we know $\hat{a}.\hat{b}=\left|\hat{a}\right|\left|\hat{b}\right|\cos (\alpha +\beta )$

$=\cos (\alpha +\beta )$

Also $\hat{a}.\hat{b}=(\cos \alpha \hat{i}+\sin \beta \hat{j}).(\cos \beta \hat{i}-\sin \beta \hat{j})$

$=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

$\therefore$ $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

proved