First slide

Trigonometric transformations

Question

If $\cos (θ - α), \cos θ, \cos (θ + α)$ are in H.P., then $\cos θ \sec (α / 2)$ is equal to

Moderate

A

$- 1$
B

$\pm \sqrt{2}$
C

$\pm 2$
D

$\pm 3$

Solution

We are given that
$\begin{aligned} \cos θ = \frac{2 \cos (θ - α) \cos (θ + α)}{\cos (θ - α) + \cos (θ + α)} \\ \Rightarrow \cos θ = \frac{2 (\cos^{2} θ - \sin^{2} α)}{2 \cos θ \cos α} \end{aligned}$
$\begin{array}{l} \Rightarrow \cos^{2} θ \cos α = \cos^{2} θ - \sin^{2} α \\ \Rightarrow \cos^{2} θ = \frac{\sin^{2} α}{1 - \cos α} \\ \Rightarrow \cos θ = \frac{4 \sin^{2} \frac{α}{2} \cos^{2} \frac{α}{2}}{2 \sin^{2} \frac{α}{2}} \\ \Rightarrow \cos^{2} θ \sec^{2} \frac{α}{2} = 2 \Rightarrow \cos θ \sec \frac{α}{2} = \pm \sqrt{2} \end{array}$

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