Question

Prove that sin theta/1-cos theta =cosec theta+ cot theta

Answer 1

TO PROVE THAT :-

$\sf\dfrac{sin\theta}{1-cos\theta}=cosec\theta+cot \theta$

SOLUTION :-

L.H.S = $\sf\dfrac{sin\theta}{1-cos\theta}$

Multiplying both numerator and denominator by $\sf 1+cos\theta$ ,

$\longrightarrow\sf\dfrac{ (sin\theta) \times (1+cos\theta)}{(1-cos\theta)\times (1+cos\theta} \\\\\longrightarrow\dfrac{ (sin\theta )\times(1+cos\theta) }{1-cos^{2}\theta}$

We know that,

$\sf sin^{2}\theta + cos^{2} \theta = 1 \\\\\therefore sin^2\theta = 1-cos^2\theta$

$\longrightarrow \sf\dfrac{(sin\theta)\times (1+cos\theta)}{sin^{2}\theta }\\\\\longrightarrow \dfrac{1+cos\theta}{sin\theta}\\\\\longrightarrow \dfrac{1}{sin\theta}+\dfrac{cos\theta}{sin\theta}\\\\\longrightarrow \bf cosec\theta+cot\theta = R.H.S$

Hence proved.