Question

If $2(\cos \theta +\sec \theta )=5$, what is the value of ${\cos }^2\theta +{\sec }^2\theta$?

• A. $\frac{25}{2}$
• B. $\frac{5}{4}$
• C. $\frac{17}{4}$
• D. $\frac{4}{17}$

Answer 1

The correct option is C $\frac{17}{4}$

$2(\cos \theta +\sec \theta )=5$

$\Rightarrow \cos \theta +\sec \theta =\frac{5}{2}$

Squaring both sides, we get

$\Rightarrow {(\cos \theta +\sec \theta )}^2=\frac{25}{4}$

$\Rightarrow {\cos }^2\theta +{\sec }^2\theta +2\cos \theta .\sec \theta =\frac{25}{4}$

$\Rightarrow {\cos }^2\theta +{\sec }^2\theta +2=\frac{25}{4}[\because \cos \theta =\frac{1}{\sec \theta }]$

$\Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{25}{4}-2$

$\Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{17}{4}$

Hence, the answer is $\frac{17}{4}.$