What Are the Solutions to sin^n(x) - cos^n(x) = 0?

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SUMMARY

The equation sin^n(x) - cos^n(x) = 0 has an infinite number of solutions, specifically at the points where tan(x) = 1. The primary solution is x = π/4 + kπ, where k is any integer. This conclusion is derived from the properties of the tangent function and its periodicity, confirming that the solutions are not limited to a single value.

PREREQUISITES
  • Understanding of trigonometric functions, particularly sine and cosine.
  • Familiarity with the tangent function and its properties.
  • Knowledge of periodic functions and their solutions.
  • Basic algebraic manipulation skills to solve equations.
NEXT STEPS
  • Study the properties of the tangent function and its periodicity.
  • Explore the implications of trigonometric identities in solving equations.
  • Learn about the general solutions of trigonometric equations.
  • Investigate the behavior of sin^n(x) and cos^n(x) for various values of n.
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Mathematics students, educators, and anyone interested in solving trigonometric equations or exploring the properties of sine and cosine functions.

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sin^n (x) - cos^n (x) = 0
 
Mathematics news on Phys.org
Is it Pi/4 ?
 
It admits an infinity of solutions,because it's simple to see that among the solutions of the initial equation,there can be found the solutions to:
tan x=1 ,x=pi/4+k*pi,k in Z

Daniel.
 

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