How can I simplify this further (or can I?)

Question

I am working through a parametrization problem and I have got to

$(ab \cos t \sin t) \sqrt{a^2 \sin^2t+b^2 \cos^2t}$

Is there any way I can simplify this using identities or other menthod?

Answer 1

If you want to integrate this expression you can substitute $u=\cos2t$ or $u=\sin^2t$ or $u=\cos^2t$ and the resulting integral is elementary.

Answer 2

It depends on that ,what are working on ...? but for example if I want to do an integral with this ,I will use : $$Y=a \sin t ,x=a \cos t \to $$ so $(ab \cos t \sin t) \sqrt{a^2 \sin^2t+b^2 \cos^2t}=xy\sqrt{x^2+y^2}$

or this idea $$(ab \cos t \sin t) \sqrt{a^2 \sin^2t+b^2 \cos^2t}=\frac{1}{2}(ab \sin 2t) \sqrt{a^2 (\frac{1-\cos 2t}{2})+b^2 (\frac{1+\cos 2t}{2})}$$