Is tan(y) = x + C, the same as Y = x/tan + C/tan?

Question

Alright so the title says it all.

I am working on a method called Separation of Variables with differential equations. The original problem was Y'= cos^2y.

I am wondering since I was trying to get "Y" all alone and move the tangent to the other side if that was acceptable or is there no way of doing it?

Answer 1

You appear to have divided each side by $\tan(x)$. Sadly, this isn't valid, for a couple of reasons:

• Less importantly, for some $x$, $\tan(x)=0$: this would amount to division by zero.

• You didn't include the argument of the tangent functions, i.e. the $(x)$ bit. This is important to be sure what your variables are.

• More importantly, you seem to have thought that

$$\tan(y)/\tan(y) = Y$$

The latter is wrong, because dividing something by itself should give you $1$.

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So what should you do?

My recommendation: take the arctangent of both sides; it is the inverse function of tangent. Thus,

$$\arctan(\tan(y)) = y$$

Then, doing this, you'd get

$$y = \arctan(x+C)$$