What is the trigonometric identity used to simplify this expression?

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SUMMARY

The discussion centers on the equivalence of the expressions \(\frac{1}{1-\sin z}\) and \(\frac{1+\sin z}{1-\sin^2 z}\). The identity used to simplify this expression is the Pythagorean identity, which states that \(1 - \sin^2 z = \cos^2 z\). Therefore, the expression can be rewritten as \(\frac{1+\sin z}{\cos^2 z}\). This confirms that the two expressions are indeed equivalent through the application of trigonometric identities.

PREREQUISITES
  • Understanding of basic trigonometric identities, specifically the Pythagorean identity.
  • Familiarity with algebraic manipulation of fractions.
  • Knowledge of sine and cosine functions.
  • Ability to simplify trigonometric expressions.
NEXT STEPS
  • Study the Pythagorean identities in trigonometry.
  • Learn how to manipulate trigonometric fractions effectively.
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  • Practice simplifying complex trigonometric expressions using various identities.
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Could anyone please tell me how are those two expressions equivalent?! what are the identities used?
 
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It's not a trig identity

[tex]\frac{1}{1-\sin z} = \frac{1+\sin z}{1+\sin z} \frac{1}{1-\sin z} = \frac{1+\sin z}{1-\sin^2 z}[/tex]
 

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