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.

Transcript

in this problem we're being asked to verify this trig identity let's go ahead and go through it so solution so when verifying trig identities we have to pick a side to start with so we have a choice we can start with the left hand side we can start with the right hand side so i'm looking at this and i haven't done it yet but i think in this case it... Read More

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.