parametric: x = cos(t), y = sin(t)

In ``Calculus'' by James Stewart, 4th edition, chapter 11, section 1 (if
I recall correctly), there's the argument that the pair of equations
above describes a circle because ~``we can eliminate t by noticing that

                  x^2 + y^2 = cos^2(t) + sin^2(t) = 1;

hence, this is a circle''~. Terminology: I use ~`` (...) ''~ to mean an
approximate quote; that is, it's coming from my memory, and hopefully
I'm not changing the meaning.

I believe that's a circle because --- so my argument goes --- a circle
of radius r has points (cos(t)/r, sin(t)/r), where r is the radius of
some circle. So if r = 1, as is the case of the unit circle, then the
pairs will be (cos(t), sin(t)). To justify, I can say that the
definition of sine (of an angle t) is opposite side of a divided by
hypothenuse, which is r. So if r = 1, sin(t) = opposite side. If the
angle t is the angle formed by the radius of the circle in the first
quadrant and the x-axis, then the opposite side is y = sin(t); reasoning
similarly for cos(t), we're done.

Stewart doesn't talk much --- for understandable reasons. But I was
hoping for an algebraic construction from the pair of the equations that
would reach

                             x^2 + y^2 = 1,

which gives me a clear elimination of t; but I only have my argument
above, if correct. 

I like algebra which doesn't need to allude to geometric pictures; it's
much safer to work this way. Any thoughts on this matter that might
improve my education will be appreciated.