Question

If \(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\) and \(\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\), then \(\cos \theta - i\sin \theta\) is equal to

• A. \(e^{i\theta}\)
• B. \(e^{-i\theta}\)
• C. \(-e^{i\theta}\)
• D. \(-e^{-i\theta}\)

Hint:

Remember the classic exponential identities:
- \( \cos \theta + i\sin \theta = e^{i\theta} \)
- \( \cos \theta - i\sin \theta = e^{-i\theta} \)
These polar-exponential identities are incredibly useful throughout complex analysis.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires simplifying a complex trigonometric expression containing exponential representations of sine and cosine functions.

Step 2: Detailed Explanation:
Let us substitute the given exponential formulas for \( \cos \theta \) and \( \sin \theta \) into the target expression:
- Given target expression:
\[ E = \cos \theta - i\sin \theta \] - Substitute the given exponential definitions:
\[ E = \left( \frac{e^{i\theta} + e^{-i\theta}}{2} \right) - i \left( \frac{e^{i\theta} - e^{-i\theta}}{2i} \right) \] Notice that the imaginary unit \( i \) in the numerator cancels out with the \( i \) in the denominator of the sine term:
\[ E = \left( \frac{e^{i\theta} + e^{-i\theta}}{2} \right) - \left( \frac{e^{i\theta} - e^{-i\theta}}{2} \right) \] Since both terms share a common denominator of 2, we can combine their numerators:
\[ E = \frac{(e^{i\theta} + e^{-i\theta}) - (e^{i\theta} - e^{-i\theta})}{2} \] Distribute the negative sign:
\[ E = \frac{e^{i\theta} + e^{-i\theta} - e^{i\theta} + e^{-i\theta}}{2} \] The positive and negative \( e^{i\theta} \) terms cancel each other out:
\[ E = \frac{2 e^{-i\theta}}{2} \] \[ E = e^{-i\theta} \] Alternatively, we can arrive at this result directly using Euler's formula:
\[ e^{-i\theta} = \cos(-\theta) + i\sin(-\theta) \] Since cosine is an even function (\( \cos(-\theta) = \cos \theta \)) and sine is an odd function (\( \sin(-\theta) = -\sin \theta \)):
\[ e^{-i\theta} = \cos \theta - i\sin \theta \] This matches Option (B).

Step 3: Final Answer:
The expression is equal to \( e^{-i\theta} \).
Therefore, the correct choice is Option (B).