Problem (1) For what value of xx, sinx+cosx=2sinx+cosx=2 Problem (2) For what value of xx, sinxcosx=1sinxcosx=1?

1 Answer
Mar 14, 2017

(sinx+cosx)(sinx+cosx) can never be 22

and sinxcosxsinxcosx can never be 11.

Explanation:

Problem (1)

sinx+cosxsinx+cosx can never be 22, the maximum value of sinx+cosxsinx+cosx can only be sqrt22 as

sinx+cosx=sqrt2(sinx xx 1/sqrt2+cosx xx 1/sqrt2)sinx+cosx=2(sinx×12+cosx×12)

= 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 sqrt22.

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.