Simplification of Complex Trigonometric Expression
Problem Statement
Simplify the following complex expression:
\[ \frac{(\cos\alpha + i\sin\alpha)^4}{(\sin\beta + i\cos\beta)^8} \]
Detailed Solution
Step 1: Express in Exponential Form Using Euler's Formula
Recall Euler's formula:
\[ \cos\theta + i\sin\theta = e^{i\theta} \]
Numerator Simplification:
\[ (\cos\alpha + i\sin\alpha)^4 = (e^{i\alpha})^4 = e^{i4\alpha} \]
Denominator Simplification:
First, rewrite the denominator:
\[ \sin\beta + i\cos\beta = i(\cos\beta - i\sin\beta) = i e^{-i\beta} \]
Then raise to the 8th power:
\[ (i e^{-i\beta})^8 = i^8 \cdot e^{-i8\beta} = 1 \cdot e^{-i8\beta} = e^{-i8\beta} \]
Step 2: Combine Numerator and Denominator
\[ \frac{e^{i4\alpha}}{e^{-i8\beta}} = e^{i4\alpha + i8\beta} = e^{i(4\alpha + 8\beta)} \]
Step 3: Convert Back to Trigonometric Form
Using Euler's formula in reverse:
\[ e^{i(4\alpha + 8\beta)} = \cos(4\alpha + 8\beta) + i\sin(4\alpha + 8\beta) \]
Final Simplified Form
\[ \boxed{\cos(4\alpha + 8\beta) + i\sin(4\alpha + 8\beta)} \]
Alternative Representation
The expression can also be written as:
\[ \cos[4(\alpha + 2\beta)] + i\sin[4(\alpha + 2\beta)] \]
This form makes the periodicity more apparent.