Question

The expression
\[ \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)} \] is equal to:

• A. \( \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right) \)
• B. \( \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right) \)
• C. \( \sin\left(\frac{\pi}{5}\right) + i \cos\left(\frac{\pi}{5}\right) \)
• D. \( \sin\left(\frac{\pi}{5}\right) - i \cos\left(\frac{\pi}{5}\right) \)
• E. \( \cos\left(\frac{\pi}{5}\right) \)

Hint:

The expression in the question is a standard form of a complex number in polar form. Simplifying complex expressions using conjugates is a helpful technique when dealing with complex fractions.

Answer 1

The Correct Option is A

The expression given is a form of complex numbers. Let’s simplify the expression: \[ \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)}. \]
This is a standard form for complex numbers, and by multiplying both the numerator and denominator by the conjugate of the denominator:
\[ \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)} \times \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}. \] The denominator becomes: \[ (1 + \cos\left(\frac{\pi}{5}\right))^2 + \sin^2\left(\frac{\pi}{5}\right) = 2(1 + \cos\left(\frac{\pi}{5}\right)). \] The numerator simplifies to: \[ 1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right), \] which is equal to: \[ \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right). \]
Thus, the final expression is \( \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right) \), which matches option (A).