This is a question that has been on my mind for sometime, and I'm getting two separate and contradictory answers to it.
If $\tan x = 1$, then what will be the value of $\sec^2 x$?
Now, one relation says that: $\tan^2 x + 1 = \sec^2 x$, which gives us $\sec^2 x = 2$.
However, if we differentiate the given equation we get a contradictory result:
$\frac{d}{dx}\tan x = \frac{d}{dx}1 \implies \sec^2 x = 0$
Is there something I'm missing? Or is it that I'm confusing two different kinds of relations which shouldn't be mixed? Can anybody help me?
$\tan x=1$ does not hold on an interval, only at a point. So the relation $\tan x=1$ cannot be differentiated. Only if a relation holds on an interval can it be differentiated to give another relation.