October 22nd, 2018, 01:27 PM  #1 
Newbie Joined: Oct 2018 From: Indoors Posts: 12 Thanks: 0  Identity crisis
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? Last edited by skipjack; October 23rd, 2018 at 07:05 AM. 
October 23rd, 2018, 07:03 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,835 Thanks: 2162 
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. Welldesigned 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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