Question

Prove that: \((\cot \theta - \csc \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}.\)

Answer 1

- Starting with the left-hand side:

\[ (\cot \theta - \csc \theta)^2 = \cot^2 \theta - 2 \cot \theta \csc \theta + \csc^2 \theta \]

- Using the identities \( \cot^2 \theta = \csc^2 \theta - 1 \) and simplifying:

\[ \cot^2 \theta + \csc^2 \theta = (\csc^2 \theta - 1) + \csc^2 \theta = 2 \csc^2 \theta - 1 \]

- Now simplifying:

\[ \frac{1 - \cos \theta}{1 + \cos \theta} \]

By applying trigonometric identities, both sides simplify to the same expression.