Verify the Trigonometric Identity 1/(1 - sin^2(x)) = 1 + tan^2(x) | Summary and Q&A

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March 29, 2021
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The Math Sorcerer
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Verify the Trigonometric Identity 1/(1 - sin^2(x)) = 1 + tan^2(x)

TL;DR

The video demonstrates how to verify a trigonometric identity by starting with the right-hand side and using addition and memorized formulas.

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Key Insights

  • 🫱 When verifying a trigonometric identity, it is important to choose a side to start with, either the left-hand side or the right-hand side.
  • 😑 Converting trigonometric functions into their equivalent forms, such as expressing tangent as sine over cosine, can simplify the expression.
  • 👻 Adding terms with the same denominator allows for further simplification.

Questions & Answers

Q: Why does the speaker choose to start with the right-hand side in verifying the trigonometric identity?

The speaker chooses to start with the right-hand side because by performing the addition, the plus sign allows for easier manipulation of the equation. Starting with the left-hand side may make it harder to simplify the expression.

Q: How is tangent expressed in terms of sine and cosine?

Tangent can be expressed as sine over cosine. Therefore, to simplify the right-hand side, the speaker writes 1 plus tangent squared as 1 plus (sine squared x over cosine squared x).

Q: How does the speaker convert 1 into a fraction with cosine squared as the denominator?

To convert 1 into a fraction with cosine squared as the denominator, the speaker writes it as cosine squared x over cosine squared x. This allows for the addition of terms by having the same denominator.

Q: What is the significance of the trigonometric identities mentioned in the video?

The trigonometric identities mentioned, such as cosine squared equals 1 minus sine squared, and sine squared equals 1 minus cosine squared, are useful formulas that can be memorized to simplify trigonometric problems. They save time and allow for quicker calculations.

Summary & Key Takeaways

  • The video explains the process of verifying a trigonometric identity by selecting one side, in this case, the right-hand side, to start with.

  • The right-hand side is simplified using the fact that tangent can be expressed as sine over cosine.

  • By converting 1 into a fraction with cosine squared as the denominator, the addition of terms can be performed.

  • The resulting expression is simplified further using the well-known trigonometric identity: cosine squared plus sine squared equals one.

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