If cosecθ = p + q p - q (p ≠ q, p ≠ 0), then |cot ((π)/(4) +...

\text{cosec} \theta = \dfrac{\text{p} + \text{q}}{\text{p} - \text{q}} ⇒ \dfrac{1}{\text\sin \theta} = \dfrac{\text{p} + \text{q}}{\text{p} - \text{q}}

let \text\tan \left(\dfrac{\theta}{2}\right) = \text{t} \text{then} \dfrac{1 + {\text{t}}^{2}}{\text{2t}} = \dfrac{\text{p} + \text{q}}{\text{p} - \text{q}} \left\{∵ \sin 2\theta = \dfrac{2tan\theta}{1 + \tan^{2}\theta}\right\}$</p><p>Using componendo and dividendo rule, we get<br /><br />$⇒ \dfrac{1 + {\text{t}}^{2} + \text{2t}}{1 + {\text{t}}^{2} - \text{2t}} = \dfrac{\text{p} + \text{q} + \text{p} - \text{q}}{\text{p} + \text{q} - \text{p} + \text{q}}$<br /><br />$⇒ {\left(\dfrac{1 + \text{t}}{1 - \text{t}}\right)}^{2} = \dfrac{\text{p}}{\text{q}}$<br /><br />$⇒ | \dfrac{1 - t}{1 + t} | = \sqrt{\dfrac{q}{p}} = | \cot ( \dfrac{\pi}{4} + \theta ) |$\left\{∵ \tan \left(\dfrac{\pi}{4} + \theta\right) = \dfrac{1 + tan\theta}{1 - tan\theta}\right\}$