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I am aware that we can gain extraneous solutions during irreversible steps such as squaring, taking tan both sides. Can we also lose some solutions?

MrSwey
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    What if both sides of the equation are equal to an angle for which the $\tan( )$ function is undefined? – JonathanZ May 26 '23 at 03:43
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    Define "irreversible step". Start for example with the equation $x^2=x$, then take (what could be called) the irreversible step of canceling $x$ on the two sides. You end up with $x=1$, and lost the solution $x=0$. – dxiv May 26 '23 at 03:46
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    You should check out the Wikipedia article on extraneous and missing solutions. To add to the examples, if squaring can introduce extraneous solutions, then taking the square root can omit some solutions. There is a simple example of this on that page. – Charlie May 26 '23 at 03:49
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    @JonathanZsupportsMonicaC I understand! For example, taking tan both sides of x = π - x doesn't give π/2. So, taking sin both sides won't lose a solution right? – MrSwey May 26 '23 at 05:58
  • Yes, sin is (first) a function, and (second) is defined on its domain of all real numbers. So the single step of applying the sin will map the quantities on both sides of the equals sign to equal values. Just make sure the sin's don't hide some solution-losing step later on, like dividing by zero. – JonathanZ May 26 '23 at 06:08
  • You should also make sure you understand @dxiv's example. Dividing by expressions that go to zero at some values is a great way to lose solutions. I personally can remember a problem where I divided out a $\sin(\theta)$ in the middle of a long problem, and wasted 30 minutes tracking it down. – JonathanZ May 26 '23 at 06:16
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    I would like to "rescue" this question. Not seen anything like it on this website, it's linked to the general concept of missing solutions which is in contrast to extraneous solutions. There's not much discussion on this on MSE either. – Sarvesh Ravichandran Iyer May 26 '23 at 14:23
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    I answered your exact question right here: "You are correct that taking $\tan$ on both sides of the equation created that extraneous solution due to $\tan$'s lack of injectivity. Such a step is actually valid even though it potentially creates extraneous solutions; importantly, it never discards solutions." In general, no ⟹ step that is valid can discard any solution. $\quad$ I've also just answered a related question: Reversible and irreversible operations in elementary algebra. @SarveshR – ryang May 27 '23 at 17:30

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Indeed, a classic false implication:

$$2x=x^2\implies 2=x$$

where you are implicitly dividing by $0$.

But if you apply a function that is defined everywhere this cannot happen as by definition $a=b$ implies $f(a)=f(b)$.

b00n heT
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