4. Zadatak

Kvadrirajmo dane jednakosti:

$$\begin{aligned}a^2&=\cos^2\alpha+\cos^2\beta+2\cos\alpha\cos\beta,\\b^2&=\sin^2\alpha+\sin^2\beta+2\sin\alpha\sin\beta.\end{aligned}$$

Zbrajanjem, odnosno oduzimanjem ovih jednakosti dobivamo:

$$\begin{aligned}a^2+b^2&=2+2(\cos\alpha\cos\beta+\sin\alpha\sin\beta)\\&=2+2\cos(\alpha-\beta),\end{aligned}$$ $$\begin{aligned}a^2-b^2&=\cos^2\alpha-\sin^2\alpha+\cos^2\beta-\sin^2\beta+2(\cos\alpha\cos\beta-\sin\alpha\sin\beta)\\&=\cos2\alpha+\cos2\beta+2\cos(\alpha+\beta)\\&=2\cos(\alpha+\beta)\cos(\alpha-\beta)+2\cos(\alpha+\beta)\\&=2\cos(\alpha+\beta)[\cos(\alpha-\beta)+1].\end{aligned}$$

Budući je $\cos(\alpha-\beta)+1=\dfrac{a^2+b^2}2\ne0$, konačno dobivamo

$$\cos(\alpha+\beta)=\frac{a^2-b^2}{a^2+b^2}.$$