Question

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

Answer 1

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.

Answer 2

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, $sinx+cosx=1$