Question

If (cos θ + sec θ) = $\frac{5}{2}$ then (cos² θ + sec² θ) = ?
(a) $\frac{17}{4}$

(b) $\frac{21}{4}$

(c) $\frac{29}{4}$

(d) $\frac{33}{4}$

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\cos \theta +\sec \theta =\frac{5}{2} \\ \cos \theta +\sec \theta =\frac{5}{2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow {\left(\cos \theta +\sec \theta \right)}^2={\left(\frac{5}{2}\right)}^2 \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta +2\left(\cos \theta \right)\left(\sec \theta \right)=\frac{25}{4} \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta +2\left(\cos \theta \right)\left(\frac{1}{\cos \theta }\right)=\frac{25}{4}\left(\because \sec \theta =\frac{1}{\cos \theta }\right) \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta +2=\frac{25}{4} \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{25}{4}-2 \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{25-8}{4} \\ \Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{17}{4} \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{a}\right). \end{gathered}$$