# Is it legal to put $\tan$ on both sides of the equation?

I have this equation and I need to find $x$ variable:

$$1+\frac{\pi}{4}-x=\arctan x$$

can I put $\tan$ on RHS and LHS in order to find $x$:

$$\tan( 1+\frac{\pi}{4}-x)=\tan(\arctan x)$$

Secondly, after I have $x$ on RHS, how am I supposed to find $x$ if the LHS has now become a complicated expression?

• Applying a function equally to both sides of an equation is always “legal”. – abiessu Jun 24 '18 at 13:20
• **implementable – user6394019 Jun 24 '18 at 13:23
• I'm afraid that the law forbids putting people in the sun – MalayTheDynamo Jun 24 '18 at 14:01
• @abiessu: But if it's a periodic function, the resulting equation might have solutions that the original equation didn't. (Michael's answer already makes this point.) The extreme case is multiplying both sides by 0, which discards most of the information from the original equation. – Peter Cordes Jun 24 '18 at 15:42
• IANAL, but it may depend on the jurisdiction. Please add more details. – tomasz Jun 24 '18 at 17:27

It is certainly true that if $1+\dfrac\pi4-x=\arctan x$ then $\tan\left(1+\dfrac\pi4-x\right)=\tan(\arctan x).$

But it is not true that if $\tan\left(1+\dfrac\pi4-x\right)=\tan(\arctan x)$ then $1+\dfrac\pi4-x=\arctan x,$ since $\tan$ is not a one-to-one function.

In other words, some of the solutions of the latter equation are not solutions of the former, so you'll need to check for extraneous roots.

You are correct, but don't think this is a good strategy. Instead consider the function $$f(x)=\arctan x+x-1-\frac{\pi}{4}$$ and show that $x=1$ is the unique zero. Try to prove that $f$ is strictly increasing by considering its derivative, or just noting that $f$ is the sum of $\arctan x$ and $x$ (which are strictly increasing) plus a constant. This is more a calculus problem than a trigonometry one!

• Considering the derivative is an overkill here since $f(x)$ is just a sum of two clearly increasing functions and a constant. – Adayah Jun 24 '18 at 15:33
• @Adayah Yes, I agree with you. Thanks for pointing out! – Robert Z Jun 24 '18 at 15:44

One obvious solution is following: x = 1.

If you plot the graphs, it will be obvious, that except x = 1 there are no other solutions.

For any arbitrary constants there is no analytical solution. Instead, solutions are based on some iterating converging process until you reach needed precision.

Yes, it is legal to apply tan to both parts. When two values are equal, then their tan values are equal, too. But if you are going to use that to get an equation that is easier to solve, pay attention, because another equation can have more solutions than your original equation and you will have to filter them to find out the solution of your original equation.