When can you divide both sides of an equation by a trigonometric function??

Question

I was told to never divide both sides of an equation by a trigonometric function in case I accidentally eliminate an answer (or divide by $0$).

However, why do we divide both sides by $\cos x$ for $\sin x = \cos x$. How do I know whether I can divide or not by a trigonometric function?

Thank you.

Answer 1

when you divide by $\cos(x)$ then it must be $$x\ne (2k+1)\frac{\pi}{2}$$ and you have to solve the equation $$\tan(x)=1$$ in an aother way you can square both sides, and we have $$\sin^2(x)=\cos^2(x)$$ or $$\sin^2(x)=1-\sin^2(x)$$ and you must check the Solutions.

Answer 2

To "never" divide by a trig function is, depending on the field, an extremly limiting advice. I often use equations and move trig functions from one side to the other by dividing both sides. As far as I remember some proofs for trig identies can also be made by explicitly using the idea that e. g. $tan(x) = \frac{sin(x)} {cos(x)}.$

The advice I got was : Whenever you transform any equation you have to have an eye on values that suddenly become possible or impossible but were not so in the first place. As also ypu have to check if you have erased values that you accidentally erased. That problem also appears in taking even roots or raising to even powers, as well as when dividing by any function that has the 0 in its range. After using those and transforming your equation, you have to check whether this is important to your desired results or if you even caused a mathematical error.

Whenever you can suggest that your function is not including zero in its outcome you can divide by it. But if it is containing a zero in its outcome then this does not have to be (it might be one!) an error but the new equation will be not defined at that certain point or crash.

Bottom line: Have an eye on what you do and check your results. That's not a mathematical well made rule, but accurate for each situation this appears. Everytime any of your functions has a 0 outcome it's restricted. When you divide by sin or cos then the resulting equation has restrictions.

@Dr. Sonnhard Graubner has pointed out that your domain is restricted and how it is restricted.

Answer 3

There are points which satisfy $\cos x=0$, but for these $x$ we have either $\sin x=1$ or $\sin x=-1$, so no such number $x$ can be a solution of the equation $\sin x=\cos x$. That's why it's safe to divide by $\cos x$ in this case (but you are right to worry about it, and it should be pointed out when you write down the steps of your solution).