Problem (1)
sinx+cosxsinx+cosx can never be 22, the maximum value of sinx+cosxsinx+cosx can only be sqrt2√2 as
sinx+cosx=sqrt2(sinx xx 1/sqrt2+cosx xx 1/sqrt2)sinx+cosx=√2(sinx×1√2+cosx×1√2)
= sqrt2(sinxcos(pi/4)+cosxsin(pi/4))√2(sinxcos(π4)+cosxsin(π4))
= sqrt2sin(x+pi/4)√2sin(x+π4)
as maximum value of any sine ratio can only be 11,
maximum value of sqrt2sin(x+pi/4)√2sin(x+π4) or sinx+cosxsinx+cosx can only be sqrt2√2.
Hence sinx+cosxsinx+cosx can never be 22.
Problem (2)
As sinxcosx=1/2(2sinxcosx)=1/2xxsin2xsinxcosx=12(2sinxcosx)=12×sin2x
and again as maximum value of any sine ratio can only be 11,
maximum value of sinxcosxsinxcosx or 1/2sin2x12sin2x can only be 1/212
and it can never be more than 1/212 and hence cannot be 11.