Question : If $\cos x+\sec x=\frac{7}{2 \sqrt{3}}$, then the value of $\cos ^2 x+\sec ^2 x$ will be_____.
Option 1: $\frac{15}{12}$
Option 2: $\frac{10}{12}$
Option 3: $\frac{25}{10}$
Option 4: $\frac{25}{12}$
Correct Answer: $\frac{25}{12}$
Solution :
Given: $\cos x+\sec x=\frac{7}{2 \sqrt{3}}$
Squaring both sides,
$(\cos x+\sec x)^2=(\frac{7}{2 \sqrt{3}})^2$
⇒ $\cos^2 x+\sec^2 x +2= \frac{49}{12}$ (Since $\cos x×\sec x=1)$
⇒ $\cos^2 x+\sec^2 x = \frac{49}{12}-2$
⇒ $\cos^2 x+\sec^2 x= \frac{25}{12}$
Hence, the correct answer is $\frac{25}{12}$.
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