Question

sin^2 theta (1/1-cos theta + 1/1+cos theta) = 2​

Answer 1

TO PROVE :–

$\\ \implies \bf \sin^2(\theta) \left( \dfrac{1}{1- \cos \theta} +\dfrac{1}{1 + \cos \theta}\right) = 2$

SOLUTION :–

• Let's take L.H.S. –

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1}{1- \cos \theta} +\dfrac{1}{1 + \cos \theta}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1 + \cos \theta +1 - \cos \theta}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1+1}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{(1)^{2} - (\cos \theta)^{2}}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{1-\cos^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{\sin^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: 2 \left( \dfrac{\sin^2\theta}{\sin^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: 2(1)$

$\\ \ \bf \: \: = \: \: 2$

$\\ \ \bf \: \: = \: \: R.H.S.$

$\\ \ \bf \: \: \longrightarrow\: \: Hence \: \: Proved$

Answer 2

TO PROVE :–

$\\ \implies \bf \sin^2(\theta) \left( \dfrac{1}{1- \cos \theta} +\dfrac{1}{1 + \cos \theta}\right) = 2$

SOLUTION :–

• Let's take L.H.S. –

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1}{1- \cos \theta} +\dfrac{1}{1 + \cos \theta}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1 + \cos \theta +1 - \cos \theta}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{1+1}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{(1- \cos \theta)(1 + \cos \theta)}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{(1)^{2} - (\cos \theta)^{2}}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{1-\cos^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: \sin^2(\theta) \left( \dfrac{2}{\sin^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: 2 \left( \dfrac{\sin^2\theta}{\sin^{2} \theta}\right)$

$\\ \ \bf \: \: = \: \: 2(1)$

$\\ \ \bf \: \: = \: \: 2$

$\\ \ \bf \: \: = \: \: R.H.S.$

$\\ \ \bf \: \: \longrightarrow\: \: Hence \: \: Proved$