Question

If $\cos \theta +\sec \theta =\frac{5}{2}$, then ${\cos }^2\theta +{\sec }^2\theta =$

• A. $\frac{21}{4}$
• B. $\frac{17}{4}$
• C. $\frac{29}{4}$
• D. $\frac{33}{4}$

Answer 1

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

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

$\Rightarrow (\cos \theta +\sec \theta {)}^2={\left(\frac{5}{2}\right)}^2$, square both sides
$\Rightarrow {\cos }^2\theta +2\cos \theta \cdot \sec \theta +{\sec }^2\theta =\frac{25}{4}$
$\Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{25}{4}-2(\because \cos \theta \cdot \sec \theta =1)$
$\Rightarrow {\cos }^2\theta +{\sec }^2\theta =\frac{17}{4}$.