Question

show that (cosecA-cotA
)2=1-cosA/1+cosA

Answer 1

Solution :-

$\tt (cosecA - cotA)^{2} = \dfrac{1 - cosA}{1 + cosA}$

Consider RHs

$\tt = (cosecA - cotA)^{2}$

Writing cosecA and cotA in terms of cosA and sinA

$\tt = \bigg( \dfrac{1}{sinA} - \dfrac{cosA}{sinA} \bigg)^{2}$

$\tt = \bigg( \dfrac{1 - cosA}{sinA} \bigg)^{2}$

$\tt = \dfrac{(1 - cosA) ^{2} }{sin^{2} A}$

$\tt = \dfrac{(1 - cosA) ^{2} }{1 - cos^{2} A}$

[ Because sin²A = 1 - cos²A ]

It can be written as

$\tt = \dfrac{(1 - cosA) ^{2} }{ 1^{2} - cos^{2} A}$

$\tt = \dfrac{(1 - cosA)^{2} }{(1 + cos A)(1 - cosA)}$

[ Because x² - y² = (x + y)(x - y) ]

$\tt = \dfrac{1 - cosA }{1 + cos A}$

= RHS

LHS = RHS

Hence proved.

Answer 2

Answer:

Refer to the attachment.