Question

can anyone help me with the 11 one? ​

Answer 1

$\underline{\texttt{Proof :}}$

$\mathrm{Now,\:\sqrt{\frac{1-cos\theta}{1+cos\theta}}}$

$\mathrm{=\sqrt{\frac{(1-cos\theta)(1-cos\theta)}{(1+cos\theta)(1-cos\theta)}}}$

$\mathrm{=\sqrt{\frac{1-2\:cos\theta+cos^{2}\theta}{1-cos^{2}\theta}}}$

$\mathrm{=\sqrt{\frac{1-2\:cos\theta+cos^{2}\theta}{sin^{2}\theta}}}$

$\mathrm{=\sqrt{\frac{1}{sin^{2}\theta}-\frac{2\:cos\theta}{sin^{2}\theta}+\frac{cos^{2}\theta}{sin^{2}\theta}}}$

$\mathrm{=\sqrt{cosec^{2}\theta-2\:cosec\theta\:cot\theta+cot^{2}\theta}}$

$\mathrm{=\sqrt{(cosec\theta-cot\theta)^{2}}}$

$\mathrm{=cosec\theta-cot\theta}$

$\to \boxed{\mathrm{\sqrt{\frac{1-cos\theta}{1+cos\theta}}=cosec\theta-cot\theta}}$

$\texttt{Hence, proved.}$