Verifying trig identities.... what about when tan is undefined?

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SUMMARY

The discussion centers on verifying the trigonometric identity tan(x)cos(x) = sin(x) and its validity at x = π/2. It is established that this identity holds true only for values of x where the left-hand side is defined, specifically excluding points where tan(x) is undefined, such as x = π/2. The conversation highlights that trigonometric identities can have restricted domains, and as x approaches π/2, the limit of tan(x)cos(x) approaches sin(π/2), suggesting continuity can extend the identity's validity.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Knowledge of limits and continuity in calculus
  • Familiarity with the unit circle and angle measures in radians
  • Basic algebraic manipulation of trigonometric identities
NEXT STEPS
  • Study the concept of limits in calculus, particularly around points of discontinuity
  • Explore the properties of trigonometric functions, focusing on their domains
  • Learn about the continuity of functions and how it applies to trigonometric identities
  • Investigate other trigonometric identities and their restrictions, such as sec(x) and cot(x)
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Students of mathematics, particularly those studying trigonometry and calculus, as well as educators seeking to clarify the nuances of trigonometric identities and their domains.

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TL;DR
Do trig identities have restricted domains?
Hello,

If I wanted to verify tan(x)cos(x) = sin(x), what about when x is pi/2? LHS has a restricted domain so it can't equal sin(x). Does this equation only work with a restricted domain? Nothing in my textbook discusses that.

Thank you
 
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You can verify that tan(x)cos(x) = sin(x) for all x where the left side is defined. That's as good as it gets.
 
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Here's a similar question. Is x/x=1? There's nothing special about trig identities potentially having holes in the domain.

One thing that is true is that as x approaches pi/2, tan(x)cos(x) approaches sin(pi/2), so if you're willing to extend the function continuously, then you get equality everywhere
 

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