$\therefore \,2\,\sec \,\theta \, = \,\sec \,(\theta - \phi ) + \sec \,(\theta + \phi )$
$ \Rightarrow \frac{2}{{\cos \theta \,}} = \frac{{\cos \,(\theta + \phi ) + \,\cos \,(\theta - \phi )}}{{\cos \,(\theta - \phi )\,\cos \,(\theta + \phi )}}$
$ \Rightarrow \,2({\cos ^2}\theta - {\sin ^2} \phi )\, = \,\cos \,\theta \,[2\,\cos \theta \,\cos \phi ]$
$ \Rightarrow \,{\cos ^2}\theta \,(1 - \cos \phi )\, = \,{\sin ^2}\phi \, = \,1 - {\cos ^2}\phi $
$ \Rightarrow \,{\cos ^2}\theta \, = 1 + \cos \phi \, = 2{\cos ^2}\frac{\phi }{2}$
$\therefore \,\,\cos \,\theta \, = \, \pm \,\sqrt 2 \cos \frac{\phi }{2}$
But given $\cos \,\theta \, = k\cos \frac{\phi }{2}$
$\therefore \,\,k = \,\, \pm \,\sqrt 2 $
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