Question

$($sin $x-\cos x{)}^2+{\cos }^2(\frac{\pi }{4}+X)\in$

• A. $[0,1$
• B. [0,2]
• C. [1,2]
• D. [0,3]

Answer 1

The correct option is D [0,3]

$f\left(x\right)=(sinx-cosx{)}^2+co{s}^2(\frac{\pi }{4}+x)$

$=1-2sinxcosx+co{s}^2(\frac{\pi }{4}+x)$

$=1-sin2x+\frac{1+cos(\frac{\pi }{2}+2x)}{2}$
$=\frac{2-2sin2x+1-sin2x}{2}[2co{s}^{2}x-1=cos2x]$
$=\frac{3-3sin2x}{2}$
$f\left(x\right)=\frac{3}{2}(1-sin2x)$

Range of $sin2xϵ[-1,1]$

Range of $f\left(x\right)ϵ[0,3]$