Question

Prove the identity :sinA /1+cosA=cosecA-cotA

Answer 1

$\large \underline{ \red{ \boxed{ \bf \green{Question}}}}$

Prove the identity :sinA /1+cosA=cosecA-cotA

$\large \underline{ \red{ \boxed{ \bf \green{Answer}}}}$

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Answer 2

${ \bold{ \green{ \underline{ANSWER} : - }}} \\ \\ \\ { \bold{ \underline{ To \: \: prove } : - }} \\ \\ { \bold{ \: \implies \: \: \frac{ \sin(A) }{1 + \cos(A) } = cosec(A) - \cot(A) }} \\ \\ \\ { \bold{ \red{ \mathfrak{ \huge{\underline{solution} : - }}}}} \\ \\ { \bold{ \: \: \: \: \: . \: \:Let's \: \: take \: \:L.H.S. - }} \\ \\ \\ { \bold{ \: \: \: { L.H.S. =\frac{ \sin(A) }{1 + \cos(A) }}}} \\ \\ { \bold{ \: \: \: \: =\frac{ \sin(A) }{1 + \cos(A) }}} \\ \\ { \bold{ \: \: \: \: . \: \: now \: \: \: rationalization - }} \\ \\ { \bold{ \: \: \: \: = \: \:\frac{ \sin(A) }{1 + \cos(A) } \times \frac{1 -\cos(A)}{1 - \cos(A)} }} \\ \\ { \bold{ \: \: \: \: = \: \: \frac{ \sin(A) \times (1 - \cos (A) )}{1 - {cos}^{2}(A) } }} \\ \\ { \bold{ \: \: \: \: = \: \: \frac{\sin(A) \times (1 - cos A)}{ {sin}^{2}(A) } }} \\ \\ { \bold{ \: \: \: \: = \: \: \frac{1 - \cos(A) }{ \sin(A) } }} \\ \\ { \bold{ \: \: \: \: = \: \: \frac{1}{ \sin(A) } - \frac{ \cos(A) }{ \sin(A) } }} \\ \\ { \bold{ \: \: \: \: = \: \: cosec( A) - \cot(A) }} \\ \\ { \bold{ \: \: \: \: = \: \: \:R.H.S. \: \: \: \: \: \: }} \\ \\ { \bold{ \huge{ \red{\boxed{ \mathtt{Hence \: \: proved}}}} }}\\ \\ \\ \\ { \bold{ \underline{ \red{ \huge{used \: \: formula} }}} : - } \\ \\ \\ { \bold{ \blue{ \: \: (1) \: (a + b)(a - b) = {a}^{2} - {b}^{2} }}} \\ \\ \\ { \bold{ \blue{ \: \: (2) \: \: 1 - {cos}^{2} (\alpha ) = {sin}^{2}( \alpha ) }}}$