October 24th, 2018, 02:35 PM  #1 
Newbie Joined: Oct 2018 From: Indoors Posts: 12 Thanks: 0  Identity crisis 2
((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. 
October 24th, 2018, 04:54 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,852 Thanks: 743 
What is your question?

October 25th, 2018, 03:00 AM  #3 
Newbie Joined: Oct 2018 From: Indoors Posts: 12 Thanks: 0 
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?

October 25th, 2018, 04:37 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,110 Thanks: 2326 
Supposed to be an identity according to what source?

October 25th, 2018, 11:47 AM  #5 
Newbie Joined: Oct 2018 From: Indoors Posts: 12 Thanks: 0 
A textbook.

October 25th, 2018, 01:21 PM  #6 
Newbie Joined: Oct 2018 From: Indoors Posts: 12 Thanks: 0  Something just dawned on me upon reading this question. While it is true that the first expression was rewritten in terms of cos, it may not be necessarily true that they are equal or identities as indicated by their graphs. https://www.wolframalpha.com/input/?...)%5E(3%2F2)%3F 
October 25th, 2018, 01:54 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 21,110 Thanks: 2326 
I wouldn't call x = ±x an identity. Was the textbook written by mathematics teachers?

October 25th, 2018, 01:54 PM  #8 
Global Moderator Joined: May 2007 Posts: 6,852 Thanks: 743 
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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