Question

\(\cot^2\left(\frac{\pi}{4} + \frac{\theta}{2}\right)\) is equal to

• A. \(\frac{1-\sin\theta}{1+\sin\theta}\)
• B. \(\frac{1-\cos\theta}{1+\cos\theta}\)
• C. \(\frac{1+\sin\theta}{1-\sin\theta}\)
• D. \(\frac{2-\sin\theta}{2+\sin\theta}\)

Hint:

Use half-angle identities for expressions like \(\frac{\pi}{4}+\frac{\theta}{2}\).

Answer 1

The Correct Option is A

Concept: \[ \tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \frac{1+\sin\theta}{\cos\theta} \]

Step 1: Invert.
\[ \cot\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \frac{\cos\theta}{1+\sin\theta} \]

Step 2: Square.
\[ = \frac{\cos^2\theta}{(1+\sin\theta)^2} \] \[ = \frac{1-\sin^2\theta}{(1+\sin\theta)^2} = \frac{1-\sin\theta}{1+\sin\theta} \]