If x = 2cos t + 3sin t and y = 2sin t - 3cos t, find the value of $x^{2} + y^{2}$
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Solution

x = 2cos t + 3sin t and
y = 2sin t - 3cos t.
We will square and add them.
$x^{2} + y^{2} = (2 c o s t + 3 s i n t)^{2} + (2 s i n t - 3 c o s t)^{2}$
= $4 c o s^{2} t + 9 s i n^{2} t + 12 s i n t c o s t + 4 s i n^{2} t + 9 c o s^{2} t - 12 s i n t c o s t$
= $4 (s i n^{2} t + c o s^{2} t) + 9 (s i n^{2} t + c o s^{2} t)$
= 13.

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