Question

The most general values of x for which
$\sin x+\cos x=mi{n}_{aϵR}\{1,a^2-4a+6\}$
are given by

• A. $2n\pi ;n\in z$
• B. $2n\pi +\frac{\pi }{2};n\in z$
• C. $n\pi +{(-1)}^n.\frac{\pi }{4}-\frac{\pi }{4};n\in z$
• D. $2n\pi -\frac{\pi }{2};n\in z$

Answer 1

The correct option is C $n\pi +{(-1)}^n.\frac{\pi }{4}-\frac{\pi }{4};n\in z$

$\sin x+\cos x=mi{n}_{aϵR}\{1,a^2-4a+6\}$
Clearly, $\sin x+\cos x=1$
$\sin (x+\frac{\pi }{4})=\frac{1}{\sqrt{2}}$
$\sin (x+\frac{\pi }{4})=\sin \frac{\pi }{4}$
$x+\frac{\pi }{4}=n\pi +{(-1)}^n\frac{\pi }{4}$
$x=n\pi +{(-1)}^n\frac{\pi }{4}-\frac{\pi }{4}$