Question

\[\frac{1 - \text{sin} \theta + \text{cos} \theta}{1 - \text{sin} \theta - \text{cos} \theta}\] = ?

Options

• - cot \[\frac{\theta}{2}\]

• cot \[\frac{\theta}{2}\]

• tan \[\frac{\theta}{2}\]

• - tan \[\frac{\theta}{2}\]

Answer 1

- cot \[\frac{\theta}{2}\]

Explanation:

We have, \[\frac{1 - \text{sin} \theta + \text{cos} \theta}{1 - \text{sin} \theta - \text{cos} \theta}\]

= \[\frac{1 - \text{cos} \theta + \text{sin} \theta}{1 - \text{cos} \theta - \text{sin} \theta}\]

`= (2 cos^2  theta/2 - 2 sin  theta/2 cos  theta/2)/(2 sin^2  theta/2 - 2 sin  theta/2 cos  theta/2)`

`= (2 cos  theta/2 (cos  theta/2 - sin  theta/2))/(2 sin  theta/2 (sin  theta/2 - cos  theta/2))`

= `(- cos  theta/2)/(sin  theta/2)`

= `- cot  theta/2`