If you input the trig identity:
$$\cot (x)+\tan(x)=\csc(x)\sec(x)$$
Into WolframAlpha, it gives the following proof:
Expand into basic trigonometric parts:
$$\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} \stackrel{?}{=} \frac{1}{\sin(x)\cos(x)}$$
Put over a common denominator:
$$\frac{\cos^2(x)+\sin^2(x)}{\cos(x)\sin(x)} \stackrel{?}{=} \frac{1}{\sin(x)\cos(x)}$$
Use the Pythagorean identity $\cos^2(x)+\sin^2(x)=1$:
$$\frac{1}{\sin(x)\cos(x)} \stackrel{?}{=} \frac{1}{\sin(x)\cos(x)}$$
And finally simplify into
$$1\stackrel{?}{=} 1$$
The left and right side are identical, so the identity has been verified.
However, I take some issue with this. All this is doing is manipulating a statement that we don't know the veracity of into a true statement. And I've learned that any false statement can prove any true statement, so if this identity was wrong you could also reduce it to a true statement.
Obviously, this proof can be easily adapted into a proof by simply manipulating one side into the other, but:
Is this proof correct on its own? And can the steps WolframAlpha takes be justified, or is it completely wrong?
Best Answer
It is good that you are wary of proving identities this way. Indeed, I could "prove" $0=1$ by saying
\begin{align*} 0 &\stackrel{?}{=}1\\ 0\cdot 0 &\stackrel{?}{=} 0 \cdot 1\\ 0 &=0. \end{align*}
The important point is that every step WolframAlpha did is reversible, while the step I took (multiplying by $0$) was not. That is what allows the proof from WolframAlpha to be rearranged into a proof that starts with one side of the identity and ends at the other:
\begin{align*} \cot(x)+\tan(x) &= \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)}\\ &= \frac{\cos^2(x)}{\sin(x)\cos(x)} + \frac{\sin^2(x)}{\sin(x)\cos(x)}\\ &= \frac{\sin^2(x)+\cos^2(x)}{\sin(x)\cos(x)}\\ &=\frac{1}{\sin(x)\cos(x)}\\ &=\csc(x)\sec(x). \end{align*}
So no, the WolframAlpha proof is not wrong, but it neglects to emphasize the important fact that every step is reversible. I am not a fan of that sort of proof, as it gives students the idea that they can prove an identity by manipulating both sides in any way they like to arrive at a true statement.