Question

Find the domain of definition of the following function.
$f\left(x\right)=\sqrt{(\sin x+\cos x{)}^2-1}.$

Answer 1

$f\left(x\right)=\sqrt{(\sin x+\cos x{)}^2-1}$

$f\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x-1}$

$f\left(x\right)=\sqrt{\sin 2x+1-1}$

$f\left(x\right)=\sqrt{\sin 2x}$

for $sin2x\ge 0$

$2xϵ[2n\pi ,(2n+1\left)\pi \right]$

so, domain will be $xϵ[n\pi ,(2n+1\left)\pi /2\right]$