Question

please help me with this sum​

Answer 1

$\mathtt{Question : \dfrac{sin\theta}{1 - cos\theta} + \dfrac{tan\theta}{1 + cos\theta}}$

$\mathtt{\implies \dfrac{sin\theta(1 + cos\theta) + tan\theta(1 - cos\theta)}{(1 - cos\theta)(1 + cos\theta)}\;\;\;(Taking\;LCM)}$

$\mathtt{\implies \dfrac{(sin\theta + sin\theta.cos\theta) + (tan\theta - tan\theta.cos\theta)}{(1 - cos^2\theta)}}$

$\mathtt{\implies \dfrac{(sin\theta + sin\theta.cos\theta) + (tan\theta - \dfrac{sin\theta}{cos\theta}.cos\theta)}{sin^2\theta}}$

$\mathtt{\implies \dfrac{(sin\theta + sin\theta.cos\theta) + (tan\theta - sin\theta)}{sin^2\theta}}$

$\mathtt{\implies \dfrac{sin\theta.cos\theta + tan\theta}{sin^2\theta}}$

$\mathtt{\implies \dfrac{sin\theta.cos\theta + \dfrac{sin\theta}{cos\theta}}{sin^2\theta}}$

$\mathtt{\implies \dfrac{sin\theta\bigg(cos\theta + \dfrac{1}{cos\theta}\bigg)}{sin^2\theta}}$

$\mathtt{\implies \dfrac{\bigg(\dfrac{cos^2\theta + 1}{cos\theta}\bigg)}{sin\theta}}$

$\mathtt{\implies {\bigg(\dfrac{cos^2\theta + 1}{sin\theta.cos\theta}\bigg)}}$

$\mathtt{\implies {\bigg(\dfrac{cos^2\theta}{sin\theta.cos\theta}\bigg) + \bigg(\dfrac{1}{sin\theta.cos\theta}\bigg)}}$

$\mathtt{\implies {\bigg(\dfrac{cos\theta}{sin\theta}\bigg) + \bigg(\dfrac{1}{sin\theta}\bigg)\bigg(\dfrac{1}{cos\theta}}\bigg)}}$

$\mathtt{\implies cot\theta + \texttt{co}\textt{sec}\theta.\texttt{sec}\theta}}$