Question

If $\sin x+\cos x=\sqrt{y+\frac{1}{y}},xϵ[0,\pi ]$ and $y>0$, then-

• A. $x=\pi /4$
• B. $x=\frac{\pi }{2}$
• C. $x=\frac{\pi }{6}$
• D. $x=3\pi /4$

Answer 1

The correct option is A $x=\pi /4$

We know that for two positive numbers $A.M\ge G.M$
$y+\frac{1}{y}\ge 2$ ...(i)
$1+sin2x=y+\frac{1}{y}$ ...(ii)
From (i) and (ii), we get
$sin2x\ge 1$
$\Rightarrow sin2x=1$
$\Rightarrow 2x=\frac{\pi }{2}$
$\Rightarrow x=\frac{\pi }{4}$
Hence, option 'A' is correct.