Question

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

Answer 1

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

and `sinxcosx` can never be `1`.

Explanation:

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`.