If $x\cos \theta + y\sin \theta = z$, then what is the value of $(x\sin \theta - y\cos \theta)^2$?

Question

Where $x$, $y$ and $z$ are variables. Answer in terms of these variables only, please.

Is there anything you personally can say about the problem? Tools that you've learned about and have considered (or even better, tried!) using, or attempts you've made that haven't worked out? — pjs36, Apr 15 '17 at 19:01
I had considered squarring both the sides but couldn't proceed further. — Khushwant Singh, Apr 15 '17 at 19:07

score 3 · Answer 1 · answered Apr 15 '17 at 19:01

$$x \cos \theta + y \sin \theta =z \implies x^2 \cos^2 \theta + y^2 \sin^2 \theta + 2 x y \sin \theta \cos \theta =z^2 \tag1$$

Let $$(x \sin \theta - y \cos \theta)^2 = x^2 \sin^2 \theta + y^2 \cos^2 \theta -2 x y \sin \theta \cos \theta = w \tag2$$

Adding $(1) $ and $(2)$

$$x^2+y^2=w+z^2 \implies (x \sin \theta - y \cos \theta)^2=x^2+y^2-z^2$$

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