Question

(1 - sin theta) (1 + sin theta) / ( 1 + cos theta) (1- cos theta)​

Answer 1

Step-by-step explanation:

(1 - sin theta) (1 + sin theta) / ( 1 + cos theta) (1- cos theta)

$(1 - \sin {}^{2} theta) \div (1 - \cos {}^{2} theta) \\ \ \\ \cos {}^{2} theta \div \sin {}^{2} theta \\ \cot {}^{2} theta$

Answer 2

$\tt \cfrac{( 1 - \sin\:\theta)(1 + \sin\:\theta)}{(1 + \cos\:\theta)(1-\cos\:\theta)}$

• (a + b)(a - b) = a² - b²

$\sf \implies \cfrac{1 - { \sin}^{2} \theta }{1 - { \cos }^{2} \theta}$

• 1 - sin² θ = cos² θ
• 1 - cos² θ = sin² θ

$\sf \implies \cfrac{ { \cos }^{2} \theta }{ \sin ^{2} \theta}$

• cos² θ/sin² θ = cot²θ

$\sf \implies { \cot }^{2} \theta$

$\underline{\boxed{\rm{\purple{\therefore \cfrac{(1 - \sin\:\theta)(1+\sin\:\theta)}{(1+\cos\:\theta)(1-\cos\:\theta)} = \cot^{2}\:\theta}}}}\:\orange{\bigstar}$