Question

If $\cos \alpha +\cos \beta =0=\sin \alpha +\sin \beta$, then prove that $\cos 2\alpha +\cos 2\beta =-2\cos \left(\alpha +\beta \right)$. [NCERT EXEMPLAR]

Answer 1

Given: $\cos \alpha +\cos \beta =0=\sin \alpha +\sin \beta$

$\therefore \cos \alpha +\cos \beta =0$

Squaring on both sides, we get

${\cos }^2\alpha +{\cos }^2\beta +2\cos \alpha \cos \beta =0.....\left(1\right)$

Also,

$\sin \alpha +\sin \beta =0$

Squaring on both sides, we get

${\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta =0.....\left(2\right)$

Subtracting (2) from (1), we get

$$\begin{gathered} \left({\cos }^2\alpha +{\cos }^2\beta +2\cos \alpha \cos \beta \right)-\left({\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta \right)=0 \\ \Rightarrow {\cos }^2\alpha -{\sin }^2\alpha +{\cos }^2\beta -{\sin }^2\beta +2\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right)=0 \\ \Rightarrow \cos 2\alpha +\cos 2\beta +2\cos \left(\alpha +\beta \right)=0 \\ \Rightarrow \cos 2\alpha +\cos 2\beta =-2\cos \left(\alpha +\beta \right) \end{gathered}$$