Question

Is it true that since sin2x+cos2x = 1, then sin(x)+cos(x) = 1? Explain your answer.

Answer 1

Answer with Step-by-step explanation:

We are given that it is true since

`sin2x+ cos2x=1` then
`sinx+cosx=1`

We have to prove that
`sinx+cosx=1`

We know that

`sin2x=2sinxcosx`

`cos2x=2cos^2x-1`

Substituting the values then we get

`2sinxcosx+2cos^2x-1=1`

`2sinxcosx+2cos^2x=1+1`

`2(sinxcosx+cos^2x)=2`

`sinxcosx+cos^2x-1=0`

`cos^2x+sinxcosx=1`

`cosx(cosx+sinx)=1`

`cosx=1,cosx+sinx=1`

Hence,
`sinx+cosx=1`

Answer 2

sin2x+cos2x=1
2sinxcosx+2cos^2x-1=1
2cosx(cosx+sinx-1)=0
2cosx=0 cosx+sinx=1
True

I did this with double angle formulas. sin2x=2sinxcosx, and cos2x=2cos^x-1. With plugging this in, I used factoring. At the end, there are two factors. One of the factors is cosx+sinx-1=0, which is cosx+sinx=1. Therefore, the answer is true.