The question is : sin ωt – cos ωt Does this function represent (a) simple harmonic (b) periodic but not simple harmonic (c) non-periodic motion?

As mentioned in the video, the equation for Simple harmonic motion is of the form A sin (ωt + Ф). Hence, if the equation has more than one term, our aim would be to simplify it and bring it to a single term.

As we know, only for sin 45 and cos 45, we get the same value 1/√2, we can think about introducing sin45 and cos 45 in the above term to bring it into a trigonometric form which will result in combining both terms to give a single sin term.

Hence, multiply and divide both term by √2, So sin ωt – cos ωt

= √2 * 1/√2 sin ωt – √2 * 1/√2 cos ωt

= √2 ( 1/√2 sin ωt –   1/√2 cos ωt )

value 1/√2 = sin 45 = cos 45 = sin ∏/4 = cos ∏/4

= √2 (cos ∏/4 sin ωt –   sin ∏/4 cos ωt )

By trigonometry formula, sin A cos B – cos A sin B = sin (A –B)

Here, A = ωt, B = ∏/4 So sin (A – B) = sin (ωt - ∏/4)

= √2 sin (ωt - ∏/4)

Comparing this with SHM equation A sin sin (ωt + Ф), we can conclude that it is SHM.