Question

If $3\tan \theta \tan \phi =1$, then $a\cos (\theta +\phi )=\cos (\theta -\phi )$
Find a

Answer 1

Given, $3\tan \theta \tan \phi =1$ or $\cot \theta \cot \phi =3$
or $\frac{\cos \theta \cos \phi }{\sin \theta \sin \phi }=\frac{3}{1}$
By componendo and dividendo, we get
$\frac{\cos \theta \cos \phi +\sin \theta \sin \phi }{\cos \theta \cos \phi -\sin \theta \sin \phi }=\frac{3+1}{3-1}$
or $\frac{\cos (\theta -\phi )}{\cos (\theta +\phi )}=2$
or $2\cos (\theta +\phi )=\cos (\theta -\phi )$
Ans: 2