Question

$\sqrt{\frac{1-\cos A}{1+\cos A}}=?$
(a) (cosec A − cot A)
(b) (cosec A + cot A)
(c) cosec A cot A
(d) None of these

Answer 1

(a) (cosec A − cot A)

$$\begin{gathered} \sqrt{\frac{1-\text{cos}A}{1+\cos A}} \\ \text{=}\sqrt{\frac{(1-\text{cos}A)}{(1+\cos A)}\times \frac{(1-\text{cos}A)}{(1-\cos A)}}\left[\begin{array}{l}\text{Multiplying the numerator and}\\ \text{denominator by}(1-\cos A)\end{array}\right] \\ \text{=}\sqrt{\frac{(1-\text{cos}A)(1-\text{cos}A)}{1-{\text{cos}}^{2}A}} \\ \text{=}\frac{1-\mathrm{co}\text{s}A}{\sqrt{1-{\text{cos}}^{2}A}} \\ \text{=}\frac{1-\mathrm{co}\text{s}A}{\sqrt{{\text{sin}}^{2}A}} \\ \text{=}\frac{1-\mathrm{co}\text{s}A}{\sin A} \\ \text{=}\frac{1}{\sin A}-\frac{\cos A}{\sin A} \\ =\cos \text{ec}A-\text{cot}A \end{gathered}$$