If-cos-cos-and-cos-are-in-HP-then-cos-sec-2-is-equal-to-a-2-b-3-c-1-2-d-None-of-these-

Question Number 16685 by rish@bh last updated on 25/Jun/17

If \cos (θ - α), \cos θ and \cos (θ + α)

are in HP, then

\cos θ \sec \frac{α}{2} is equal to

a) \pm \sqrt{2}

b) \pm \sqrt{3}

c) \pm \frac{1}{\sqrt{2}}

d) None of these

Answered by sushmitak last updated on 25/Jun/17

\cos θ = \frac{2 \cos (θ - α) \cos (θ + α)}{\cos (θ + α) + \cos (θ - α)}

\cos θ = \frac{\cos 2 θ + \cos 2 α}{2 \cos θ \cos α}

{2 \cos}^{2} θ \cos α = {2 \cos}^{2} θ - 1 + {2 \cos}^{2} α - 1

\cos^{2} θ \cos α = \cos^{2} θ + \cos^{2} α - 1

\cos^{2} θ (\cos α - 1) = (\cos α - 1) (\cos α + 1)

\frac{\cos^{2} θ}{1 + \cos α} = 1

\frac{\cos^{2} θ}{{2 \cos}^{2} \frac{α}{2}} = 1

\cos^{2} θ \sec^{2} \frac{α}{2} = 2

\cos θ \sec \frac{α}{2} = \pm \sqrt{2}