How can I prove (cos 3x - Cos 7x) / (Sin 3x + Sin 7x) = Tan 2x?

  • Context: Undergrad 
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SUMMARY

The expression (cos 3x - cos 7x) / (sin 3x + sin 7x) simplifies to -tan 2x through the application of trigonometric identities. The proof involves using the sum-to-product identities, specifically transforming the numerator and denominator into sine and cosine forms. The final steps confirm that the left-hand side equals the right-hand side, validating the equation. The key identities used include the sine and cosine of angle sums and differences.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sum-to-product identities.
  • Familiarity with the tangent function and its relationship to sine and cosine.
  • Knowledge of angle addition and subtraction formulas in trigonometry.
  • Basic algebraic manipulation skills for simplifying trigonometric expressions.
NEXT STEPS
  • Study the derivation and applications of sum-to-product identities in trigonometry.
  • Learn about angle addition and subtraction formulas for sine and cosine.
  • Explore advanced trigonometric proofs and their applications in calculus.
  • Practice simplifying complex trigonometric expressions using various identities.
USEFUL FOR

Students studying trigonometry, mathematics educators, and anyone interested in mastering trigonometric proofs and identities.

powp
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Hello

I have to prove that

(cos 3x - Cos 7x) / (Sin 3x + Sin 7x) = Tan 2x

So this is what I am doing

-2 Sin (3x + 7x)/2 Sin (3x-7x)/2
--------------------------------
2 Sin (3x + 7x)/2 Cos (3x-7x)/2

-2 Sin 5x Sin -2x
-----------------
2 Sin 5x Cos -2x

Sin 2x
-------
-cos 2x

Which equals -Tan 2x. What am i doing wrong??

Thanks
 
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-2 Sin 5x Sin -2x
-----------------
2 Sin 5x Cos -2x

Sin 2x
-------
-cos 2x

Cos(-2x)=Cos(2x)
Then i believe everything is ok.
 
Thanks for your help.
 

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