How can I simplify this step by step?

Question

I want to simplify this :

$$\frac{x^2}{a^2} + \frac{(x\tan \theta)^2}{b^2} = 1$$

The answer is supposed to be this :

$$x = \frac{ab}{\sqrt{b^2+a^2\tan^2 \theta}}$$

I’d like to understand this step by step

Answer 1

Begin by rewriting the first equation as: $$x^2\left(\frac{1}{a^2} + \frac{\tan^2\theta}{b^2} \right) = 1$$ You can then take a common denominator to get: $$x^2\left(\frac{b^2 + a^2\tan^2\theta}{a^2b^2} \right) = 1$$ Divide out to isolate $x$: $$x^2 = \frac{a^2b^2}{b^2 + a^2\tan^2\theta} $$ And finally, take the square root of both sides (actually, there should be both positive and negative solutions to this equation, as we began with $x^2$; both the positive and negative values are legal): $$x = \pm \frac{ab}{\sqrt{b^2 + a^2\tan^2\theta}} $$ As long as this is well-defined.