Trouble with a tangent series expansion

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SUMMARY

The discussion centers on expanding the function ln(cos(x)) using the Taylor series expansion of tan(x), specifically the series tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7. The user seeks assistance in identifying equivalencies and derivatives related to ln(cos(x)). It is concluded that examining the derivative of ln(cos(x)) will clarify the relationship and facilitate the expansion process.

PREREQUISITES
  • Understanding of Taylor series expansions
  • Familiarity with trigonometric functions, specifically cos(x) and tan(x)
  • Knowledge of derivatives and their applications in calculus
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the Taylor series expansion of ln(cos(x))
  • Learn about the derivatives of trigonometric functions
  • Explore the relationship between tan(x) and ln(cos(x))
  • Investigate convergence criteria for Taylor series
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Students and educators in calculus, mathematicians focusing on series expansions, and anyone interested in the application of derivatives in trigonometric functions.

iatnogpitw
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Okay, so I am supposed to expand ln(cos(x)) basing my calculations on tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + ... and expanding out to that fourth term is sufficient. I am having a major brain meltdown, I can't seem to find any equivalencies present, and I'm sure there are. Any help is greatly appreciated.
 
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Checking what the derivative of the ln(cos(x)) is, should make it evident what is happening.
 

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