Kelly Nelson

2022-01-01

Can anyone see a way to simplify one of these expressions?
${\mathrm{cos}}^{2}\theta \mathrm{sin}\varphi +{\mathrm{sin}}^{2}\theta \mathrm{cos}\varphi$
or
${\mathrm{sin}}^{2}\theta \mathrm{sin}\varphi -{\mathrm{cos}}^{2}\theta \mathrm{cos}\varphi$
Ive

encolatgehu

${\mathrm{cos}}^{2}\mathrm{sin}\varphi +{\mathrm{sin}}^{2}\theta \mathrm{cos}\varphi =$
$=\left(1-{\mathrm{sin}}^{2}\left(\theta \right)\right)\mathrm{sin}\varphi +{\mathrm{sin}}^{2}\theta \mathrm{cos}\varphi =$
$=\mathrm{sin}\varphi +{\mathrm{sin}}^{2}\left(\theta \right)\left(\mathrm{cos}\left(\varphi \right)-\mathrm{sin}\varphi \right)=$
$=\mathrm{sin}\varphi +{\mathrm{sin}}^{2}\left(\theta \right)\left(\mathrm{cos}\left(\varphi \right)+\mathrm{cos}\left(\varphi +\frac{\pi }{4}\right)\right)=$
$=\mathrm{sin}\varphi +2{\mathrm{sin}}^{2}\left(\theta \right)\left(\mathrm{cos}\left(\varphi +\frac{\pi }{4}\right)\mathrm{cos}\left(\frac{\pi }{4}\right)\right)$
$=\mathrm{sin}\varphi +\sqrt{2}{\mathrm{sin}}^{2}\left(\theta \right)\mathrm{cos}\left(\varphi +\frac{\pi }{4}\right)$

redhotdevil13l3

The expressions are pretty simple. You could write
$f\phantom{\rule{0.222em}{0ex}}={\mathrm{cos}}^{2}\theta \mathrm{sin}\varphi +{\mathrm{sin}}^{2}\theta \mathrm{cos}\varphi =\frac{1}{\sqrt{2}}\left(\mathrm{sin}\left(\frac{\pi }{4}+\varphi \right)-\mathrm{cos}2\theta \mathrm{sin}\left(\frac{\pi }{4}-\varphi \right)\right)$
$g\phantom{\rule{0.222em}{0ex}}={\mathrm{sin}}^{2}\theta \mathrm{sin}\varphi -{\mathrm{cos}}^{2}\mathrm{cos}\varphi =-\frac{1}{\sqrt{2}}\left(\mathrm{sin}\left(\frac{\pi }{4}-\varphi \right)+\mathrm{cos}2\theta \mathrm{sin}\left(\frac{\pi }{4}+\varphi \right)\right)$
but those are not what I'd call "simpler". (That said, if your context lends some significance to the quantity $\varphi ±\frac{\pi }{4}$, then there could be some benefit here.)
Depending upon your needs, it could be useful to write them as
$f={\mathrm{cos}}^{2}\theta \mathrm{cos}\varphi \left({\mathrm{tan}}^{2}\theta +\mathrm{tan}\varphi \right)$
$g=-{\mathrm{cos}}^{2}\theta \mathrm{cos}\varphi \left(1-{\mathrm{tan}}^{2}\theta \mathrm{tan}\varphi \right)$
Those may-or-may-not seem better, but note that the ratio f/g looks an awful lot like (the negative of) the angle-addition formula for tangent ... if only we had ${\mathrm{tan}}^{2}\theta =\mathrm{tan}\psi$ for some $\psi$. But in that case, we'd have
${\mathrm{cos}}^{2}\theta =\frac{1}{1+{\mathrm{tan}}^{2}\theta }=\frac{1}{1+\mathrm{tan}\psi }=\frac{\mathrm{cos}\psi }{\mathrm{sin}\psi +\mathrm{cos}\psi }=\frac{\mathrm{cos}\psi }{\sqrt{2}\mathrm{sin}\left(\psi +\frac{\pi }{4}\right)}$
${\mathrm{sin}}^{2}\theta =\frac{\mathrm{sin}\psi }{\sqrt{2}\mathrm{sin}\left(\psi +\frac{\pi }{4}\right)}$
whereupon, we'd obtain

Those are also not necessarily "simpler" than the original forms, and the $\mathrm{tan}2\theta \to \mathrm{tan}\psi$ re-parameterization may not be appropriate for your particular needs
We couuld even take it further, defining :

but there's a danger of veering too far outside the context of your investigation.

Vasquez

Hint:

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