how do i prove that the resultant voltage V3 is in the same frequency as v1 and v2 ?
The frequency of v1 is w. The frequency of v2 is also w. It is a wave that has 2/3 of amplitude of v1 and phase shifted by 90deg. v3 has the same frequency w but its amplitude is 1.2 times that of v1 and phase shifted by 33.7 degrees. You can prove mathematically by expanding v3. Then you get a sine wave and a cosine wave with frequency w each. viz: v3=3.6sinwt.cos33.7+3.6coswt.sin33.7. You can write this as asinwt+bcoswt, where a and b are constants (the amplitudes). Hope this helps.
The general equation of a sine wave (and similarly, cosine wave) is
A affects amplitude
B affects frequency
C affects horizontal shift
D affects vertical shift
Do you see that the B value for all 3 of your equations is the same variable, therefore they must all have the same frequency.