Two waves possess the same wave shape

(i) if they contain the same harmonics 

(ii) if the ratio of the corresponding harmonics to their respective fundamentals is the same 

(iii) if the harmonics are similarly spaced with respect to their fundamentals. In other words, 

(a) the ratio of the magnitudes of corresponding harmonics must be constant and 

(b) with fundamentals in phase, the corresponding harmonics of the two waves must be in phase. 

The test is applied first by checking the ratio of the corresponding harmonics and then coinciding the fundamentals by shifting one wave. If the phase angles of the corresponding harmonics are the same, then the two waves have the same shape. 

In the present case, condition (i) is fulfilled because the voltage and current waves contain the same harmonics, i.e. third and fifth. 

Secondly, the ratio of the magnitude of corresponding current and voltage harmonics is the same i.e. 1/10. 

Now, let the fundamental of the current wave be shifted ahead by 90º so that it is brought in phase with the fundamental of the voltage wave. It may be noted that the third and fifth harmonics of the current wave will be shifted by 3 × 90º = 270° and 5 × 90º = 450º respectively. Hence, the current wave becomes

i' = 1.0sin(ωt - 60º + 90º) + 5sin(3ωt - 150º + 270º) + 2.5cos(5ωt - 140º + 450º)

= 1.0sin(ωt + 30º) + 5sin(3ωt + 120º) + 2.5cos(ωt + 310º)

= 1.0sin(ωt + 30º) - 5sin(3ωt - 60º) + 2.5sin(5ωt + 40º)

It is seen that now the corresponding harmonics of the voltage and current waves are in phase. Since all conditions are fulfilled, the two waves are of the same wave shape.