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The expression \((\csc \theta - \cot \theta)^2\) is equal to:

  • A.\(\frac{1+\cos \theta}{1-\cos \theta}\)
  • B.\(\frac{1-\cos \theta}{1+\cos \theta}\)
  • C.\(\frac{1+\sin \theta}{1-\sin \theta}\)
  • D.\(\frac{1-\sin \theta}{1+\sin \theta}\)

Answer: B

Convert to sines and cosines:

\((\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta})^2 = (\frac{1 - \cos \theta}{\sin \theta})^2 = \frac{(1 - \cos \theta)^2}{\sin^2 \theta}\).

Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta)\), the expression becomes:

\(\frac{(1 - \cos \theta)^2}{(1 - \cos \theta)(1 + \cos \theta)} = \frac{1-\cos \theta}{1+\cos \theta}\).

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