Proving Trig Problem: sec (2x) - 1 = sin^2 x / 2 sec (2x)

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SUMMARY

The equation sec(2x) - 1 = sin²x / (2 sec(2x) ) can be proven using trigonometric identities. The key is to express sec(2x) as 1/cos(2x), which simplifies to 1/(cos²x - sin²x). By applying the double angle identities for sine and cosine, the left side can be transformed into terms of sine and cosine, leading to the desired equality. This approach clarifies the relationship between sec(2x) and the sine function.

PREREQUISITES
  • Understanding of trigonometric identities, specifically secant and sine functions.
  • Familiarity with double angle identities for sine and cosine.
  • Basic algebraic manipulation skills to handle fractions and equations.
  • Knowledge of how to substitute trigonometric functions in equations.
NEXT STEPS
  • Study the double angle identities for sine and cosine in detail.
  • Practice proving trigonometric identities using algebraic manipulation.
  • Explore the relationship between secant and cosine functions.
  • Learn about common trigonometric substitutions in equations.
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric identities, and anyone looking to enhance their problem-solving skills in mathematics.

Sixlets
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I'm stuck :/ I have to prove the following:
sec (2x) - 1 = sin^2 x
____________
2 sec (2x)

Unfortunately, I don't know any terms that are equal to sec (2x). Any help would be awesome!
 
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sec(2x)=\frac{1}{cos(2x)}=\frac{1}{cos^2x-sin^2x}

Also you can change the denominator by substituting in for cos^2x or sin^2x.
 
I think I would start with the left side of the eq. and use the double angle identity to get it in terms of sin/cos.
 
I'll try that again. I kept getting way off track, but I probably just got confused with all the fractions haha. Thanks everyone!
 

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