Solve 3cos^2(3x)+3sin^2(3x)=3: Trig Identities

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SUMMARY

The equation 3cos²(3x) + 3sin²(3x) = 3 is validated through the application of the fundamental trigonometric identity sin²(z) + cos²(z) = 1. By substituting 3x with z, the equation simplifies to 3[cos²(z) + sin²(z)] = 3, confirming its correctness. This identity highlights the consistency of trigonometric functions across transformations.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin² + cos² = 1
  • Familiarity with variable substitution in mathematical equations
  • Basic knowledge of trigonometric functions and their properties
  • Ability to manipulate algebraic expressions involving trigonometric terms
NEXT STEPS
  • Study advanced trigonometric identities and their proofs
  • Learn about variable substitution techniques in trigonometry
  • Explore the applications of trigonometric identities in calculus
  • Investigate the implications of trigonometric transformations in physics
USEFUL FOR

Students, educators, and mathematicians interested in deepening their understanding of trigonometric identities and their applications in various mathematical contexts.

mathguyz
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Everyone knows the obvious trig identities like sin^2 + cos^2 =1, cosx=1+ sin^2, and tanx =sin/cos. I ran across an old identity the other day: 3cos^2(3x)+3sin^2(3x)=3. Can anyone here figure out why and how? I tried it and couldn't figure it out.
 
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Look at your first "obvious trig identity"...
 
3cos^2(3x)+3sin^2(3x)=3[cos^2(3x)+sin^2(3x)] , now substitude 3x=z and using sin^2(z) + cos^2(z) =1 you have it!

NB: I´m not sure this is correct, whatever it is meant to be: cosx=1+ sin^2
 

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