Question

(Cosec A / Cosec A-1) + (Cosec A / Cosec A +1) = 2 Sec²A ​

Answer 1

Step-by-step explanation:

LHS

$\frac{ \csc (\alpha) }{ \csc (\alpha) - 1 } + \frac{ \csc (\alpha) }{ \csc (\alpha) + 1 } \\ = \frac{ \csc( \alpha )( \csc( \alpha ) + 1 ) + \csc( \alpha )( \csc( \alpha ) - 1) }{( \csc( \alpha ) - 1 )( \csc( \alpha ) + 1 )} \\ = \frac{ { \csc }^{2}( \alpha ) + \csc( \alpha ) + { \csc }^{2}( \alpha ) - \csc( \alpha ) }{ { \csc}^{2}( \alpha ) - 1 } \\ = \frac{2 { \csc }^{2} ( \alpha ) }{ { \csc }^{2}( \alpha ) - 1 } \\ = \frac{2 { \csc}^{2} ( \alpha )}{ { \csc }^{2} ( \alpha )} - 2 { \csc}^{2} ( \alpha )\\ = 2 - 2 { \csc}^{2} ( \alpha ) \\ = 2(1 - { \csc }^{2}( \alpha )) \\ = 2 { \sec }^{2}( \alpha )$

Identity used:

1 - csc²x = sec²x

SINCE LHS = RHS

HENCE PROVED