Question

What will be the resultant amplitude when the waves y1 = Asinwt and y2 = Acoswt are superimposed?

Answer 1

phase difference between the two waves = 90°
addition of amplitudes follows vector sum rule 
so the resultant will be = √(A² + A² + 2AAcos90)
                                  = √(2A²)
                                  = A√2 
 

Answer 2

Acos(wt) = Asin(90+wt) 
Therefore phase difference = 90
Since the frequency is same
Resultant Vector is = $\sqrt{ A^{2} + A^{2} } = A \sqrt{2} sin(wt)$