Question : If $(\cos q+\sin q)=\frac{31}{25}$, then what will be the value of $\cos^2 q$?
Option 1: $\frac{522}{625}$
Option 2: $\frac{512}{625}$
Option 3: $\frac{513}{625}$
Option 4: $\frac{576}{625}$
Correct Answer: $\frac{576}{625}$
Solution :
\(\cos q + \sin q = \frac{31}{25}\) ____(i)
Squaring both sides, we get,
$⇒(\cos q + \sin q)^2 = \left(\frac{31}{25}\right)^2$
$⇒\cos^2 q + \sin^2 q + 2\cos q \sin q = \left(\frac{31}{25}\right)^2$
$⇒1 + 2\cos q \sin q = \left(\frac{31}{25}\right)^2$
$⇒2\cos q \sin q = \frac{333}{625}$
Also, $(\cos q - \sin q)^2 =1 - 2\cos q \sin q$
$⇒(\cos q - \sin q)^2 =1 - \frac{333}{625}=\frac{289}{625}$
$⇒(\cos q - \sin q) =\frac{17}{25}$ ____(ii)
Adding equations (i) and (ii), we get,
$⇒2\cos q=\frac{48}{25}$
$⇒\cos q=\frac{24}{25}$
$\therefore\cos ^2q=(\frac{24}{25})^2=\frac{576}{625}$
Hence, the correct answer is $\frac{576}{625}$.
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