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?
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