Question

Consider the function f(x) = ln(sin(x)). Expand f(x+h) using Taylor's series. In this context, the correct statement(s) is/are

• A. Second term in the Taylor's series i.e., the term which includes his: h. ln(sin(x))
• B. First term is ln(sin(x))
• C. Third term in the Taylor's series i.e., the term which includes $h^2$ is: $\frac{-h^2}{2(sin(x))^2}$
• D. Third term in the Taylor's series i.e., the term which includes $h^2$ is: $\frac{2h^2}{(sin(x))^2}$

Answer 1

The Correct Option is B, C

The Taylor series expansion of \( f(x) \) about \( x \) yields: \[ f(x+h) = \ln(\sin(x)) + \cot(x)h - \frac{\csc^2(x)}{2}h^2 + \cdots \] Verifying: - The first term \( \ln(\sin(x)) \) is straightforward as the function's value at \( x \). - The third term involving \( \frac{h^2}{2} \), calculated from \( f''(x) = -\csc^2(x) \), is \( -\frac{h^2}{2\sin^2(x)} \), confirming (C).