# Identity crisis 2

#### TheReluctantMathMan

((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.

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1 person

#### TheReluctantMathMan

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
Supposed to be an identity according to what source?

A textbook.

#### skipjack

Forum Staff
I wouldn't call |x| = Â±x an identity. Was the textbook written by mathematics teachers?

#### mathman

Forum Staff
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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