Identity crisis 2

Oct 2018
12
0
Indoors
((cot (x))^2+1)/sin (x) expressed in terms of cos (x) is supposedly + or - 1/(1-(cos (x))^2)^(3/2).

The graph of the left side is like a slimmer version of the csc function and not surprisingly is tangent to the minimum and maximum points of the sine function.

The graph of 1/(1-(cos (x))^2)^(3/2) is like a slimmer version of |csc (x)| function and is tangent to the maximum points of the |sin (x)| function.

The graph of -1/(1-(cos (x))^2)^(3/2) is like a slimmer version of -|csc (x)| function and is tangent to the minimum points of the -|sin (x)| function.

As you can see, these differing graphs makes me want to reevaluate identities.
 
Oct 2018
12
0
Indoors
If I must ask a question, it would probably have something to do with how one would reconcile the difference in these graphs considering that they are supposed to be identities. Does it not bother you that something this basic is somehow inconsistent?
 

skipjack

Forum Staff
Dec 2006
21,296
2,375
Supposed to be an identity according to what source?
 
Oct 2018
12
0
Indoors

skipjack

Forum Staff
Dec 2006
21,296
2,375
I wouldn't call |x| = ±x an identity. Was the textbook written by mathematics teachers?
 

mathman

Forum Staff
May 2007
6,886
759
Any expression involving square roots has to be handled carefully taking into account the two possibilities. All three expressions are (in a sense) the same, when properly handling this ambiguity.
 
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