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prod_(i=1)^(n)((1)/(1+e^(i theta)))^(1-x_(i))((e^(i theta))/(1+e^(i theta)))^(x_(i))

∏i=1n(11+eiθ)1−xi(eiθ1+eiθ)xi \prod_{i=1}^{n}\left(\frac{1}{1+e^{i \theta}}\right)^{1-x_{i}}\left(\frac{e^{i \theta}}{1+e^{i \theta}}\right)^{x_{i}}

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Q. ∏i=1n(11+eiθ)1−xi(eiθ1+eiθ)xi \prod_{i=1}^{n}\left(\frac{1}{1+e^{i \theta}}\right)^{1-x_{i}}\left(\frac{e^{i \theta}}{1+e^{i \theta}}\right)^{x_{i}}
  1. Simplify expression using properties: Simplify the expression inside the product using properties of exponents. \newline(11+eiθ)1−xi×(eiθ1+eiθ)xi=11−xi×eiθ×xi1+eiθ\left(\frac{1}{1+e^{i \theta}}\right)^{1-x_{i}} \times \left(\frac{e^{i \theta}}{1+e^{i \theta}}\right)^{x_{i}} = \frac{1^{1-x_{i}} \times e^{i \theta \times x_{i}}}{1+e^{i \theta}}
  2. Cancel out term in numerator and denominator: Notice that the eiθ∗xie^{i \theta * x_{i}} term cancels out in the numerator and denominator.\newlineSo, the expression simplifies to (11−xi)/(1+eiθ)(1^{1-x_{i}})/(1+e^{i \theta})

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