Identity crisis

Oct 2018
12
0
Indoors
From a College Algebra and Trigonometry textbook: An identity is an equation for which the solution set is the same as the domain of the variable.

Accordingly, the equation (tan(x))^2*csc(x) = sin(x)/(1-(sin(x))^2) is supposed to be an identity with the right side expressed in terms in sin(x).

While I've accepted this at face value, it bothered me that the left side has a lesser domain than the right side. The left side is not defined at x=.5*pi+k*pi and x=k*pi where k is an integer. The right side, on the other hand, is not defined only at x=.5*pi+k*pi.

The graphs of both are, of course, nearly identical, with one minor difference - the left side has holes at x=k*pi, where k is an integer.

I have noticed this with many other "identities".
So what's going on here?
 
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skipjack

Forum Staff
Dec 2006
21,301
2,377
Many identities are given without the domain being stated explicitly. The domain is then given implicitly, but the rules for determining the implicit domain differ between authors, with some authors not stating the rules they use. Well-designed examination questions avoid this issue. However, I have once seen an examination question that asked for a differential equation to be solved, explicitly stating (by mistake) a domain that was unachievable.
 
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