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

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Answer 1 · 2017-03-14T14:44:15

$(sinx+cosx)$ can never be $2$

and $sinxcosx$ can never be $1$ .

Problem (1)

$sinx+cosx$ can never be $2$ , the maximum value of $sinx+cosx$ can only be $sqrt2$ as

$sinx+cosx=sqrt2(sinx xx 1/sqrt2+cosx xx 1/sqrt2)$

= $sqrt2(sinxcos(pi/4)+cosxsin(pi/4))$

= $sqrt2sin(x+pi/4)$

as maximum value of any sine ratio can only be $1$ ,

maximum value of $sqrt2sin(x+pi/4)$ or $sinx+cosx$ can only be $sqrt2$ .

Hence $sinx+cosx$ can never be $2$ .

Problem (2)

As $sinxcosx=1/2(2sinxcosx)=1/2xxsin2x$

and again as maximum value of any sine ratio can only be $1$ ,

maximum value of $sinxcosx$ or $1/2sin2x$ can only be $1/2$

and it can never be more than $1/2$ and hence cannot be $1$ .

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