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This is part of a question about the range of tan. The algebra is tripping me up. Let $P = (x, y)$ be the point on the unit circle that corresponds to an angle $t$. Consider the equation $\tan(t)= \frac yx = a$. Then $y = ax.$ Now $x^2 + y^2 = 1$, so $x^2+ a^2x^2 = 1$

Thus,$$x = \pm \frac{1}{\sqrt{1+a^2}}$$ And$$y = \pm \frac{a}{\sqrt{1+a^2}}$$ How do you get from $x^2+ a^2x^2 = 1$ to those two equation? All I can get is $x^2 = 1 - a^2x^2$ which doesn't go anywhere. Thanks.

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    $\begingroup$ Factor out $x^2$ in $x^2+a^2x^2 = 1$ to get $x^2(1+a^2) = 1$. Now divide through. $\endgroup$ Jan 31, 2023 at 20:41

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You can take $x^2$ out and then take square root on both sides-

$$x^2 + a^2x^2 = 1 \\ => x^2(1 + a^2) = 1 \\ => x^2= \dfrac{1}{1 + a^2} \\ \therefore x= \pm\dfrac{1}{\sqrt{1 + a^2}} \\ y = ax = \pm\dfrac{a}{\sqrt{1 + a^2}} $$

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