How to Prove the Expansion of e^(cos^(-1)(x))

TL;DR

To prove that e^(cos^(-1)(x)) equals e^(pi/2)(1 - x + x^2/2 - x^3/3 + 5x^4/24 + ...), apply the Leibnitz theorem for differentiation. Start by expressing y as e^(cos^(-1)(x)) and differentiate to find the general terms in the series expansion, confirming the identity through the process.

Transcript

hello everyone in this session we'll see another question on expansion using lebanese method so it says prove that e to the power of cos inverse of x is equal to e to the power of pi by 2 1 minus x plus x square by 2 minus 1 by 3 x cubed plus 5 by 24 x to the power 4 and so on other terms so let y is equal to e to the power of cos inverse of x and ... Read More

Key Insights

• 🍉 The Lebanese method can be used to find the general terms of an expansion.
• 🤩 Differentiation and applying Lebanese theorem are key steps in finding the expansion.
• ☺️ The expansion for e^(cos^(-1)(x)) is equal to e^(pi/2)(1 - x + x^2/2 - x^3/3 + 5x^4/24 + ...).

Questions & Answers

Q: What is the function being expanded using the Lebanese method?

The function being expanded is e^(cos^(-1)(x)).

Q: How is the Lebanese method applied to find the expansion?

The Lebanese method involves differentiating the function and applying Lebanese theorem to find the general terms of the expansion.

Q: What is the general term for the expansion?

The general term for the expansion is 1 - x + x^2/2 - x^3/3 + 5x^4/24 + ...

Q: What is the value of the function at x = 0?

The value of the function at x = 0 is e^(pi/2).

Summary & Key Takeaways

• The content explains how to use the Lebanese method to prove an expansion for the function e^(cos^(-1)(x)).

• The video walks through the process of differentiating and applying Lebanese theorem to find the general terms of the expansion.

• The expansion is proven to be equal to e^(pi/2)(1 - x + x^2/2 - x^3/3 + 5x^4/24 + ...).