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?

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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