Question

cos theta / 1 - sin theta = 1+ sin theta / cos theta​

Answer 1

       $\underline{\bold{Explaination:}}$

       $\text{The given expression is:}$

       $\boxed{\frac{cos\theta}{1-sin\theta} = \frac{1 + sin\theta}{cos\theta} }$

       $\text{Taking L.H.S}$

$\implies\boxed{\frac{cos\theta}{1-sin\theta}}$

       $\text{Multiplying and dividing by }(1+sin\theta)$

$\implies\boxed{\frac{cos\theta}{1-sin\theta}*\frac{(1+sin\theta)}{(1+sin\theta)} }$

       $\text{We know that, }(a+b)(a-b) = a^{2} - b^{2}$

$\implies\boxed{\frac{cos\theta(1+sin\theta)}{1-sin^{2} \theta}}$

       $\text{We know that, }1- sin^{2}\theta = cos^{2}\theta$

$\implies\boxed{\frac{cos\theta(1+sin\theta)}{cos^{2}\theta }}$

       $\text{Simplifying...}$

$\implies\boxed{\frac{1+sin\theta}{cos\theta }}$

$\implies \text{L.H.S = R.H.S}$

  $\therefore \text{ }\text{ Hence Proved}$