Suppose and (0) are such that (+), , (-) are in A.P.

Question

Suppose $	heta$ and $\varphi\ (
e0)$ are such that $\sec(	heta+\varphi), \sec	heta, \sec(	heta-\varphi)$ are in A.P. If $\cos	heta=k\cos\dfrac{\varphi}{2}$ for some $k$, then $k$ is equal to:

Accepted Answer

A.P. condition: $2\sec\theta=\sec(\theta-\varphi)+\sec(\theta+\varphi)$, i.e.
$$\frac{2}{\cos\theta}=\frac{\cos(\theta+\varphi)+\cos(\theta-\varphi)}{\cos(\theta-\varphi)\cos(\theta+\varphi)}=\frac{2\cos\theta\cos\varphi}{\cos^2\theta-\sin^2\varphi}.$$

$$\Rightarrow \cos^2\theta(1-\cos\varphi)=\sin^2\varphi=1-\cos^2\varphi \Rightarrow \cos^2\theta=1+\cos\varphi=2\cos^2\frac{\varphi}{2}.$$

$$\cos\theta=\pm\sqrt2\cos\frac{\varphi}{2} \Rightarrow k=\pm\sqrt2.$$

Solutions · 1