Question

cosec A/cosec-1+cosecA/cosecA+1=2secA​

Answer 1

Correct question:

$\\ \sf \implies \boxed{\sf \dfrac{ cosec \: A }{cosec \: A - 1} + \dfrac{ cosec \: A }{cosec \: A + 1} = 2 \sec^{2} A}$

Solving LHS,

$\\ \sf \implies {\sf \dfrac{ \frac{1}{ \sin A} }{\frac{1}{\sin A} - 1} + {\sf \dfrac{ \frac{1}{ \sin A} }{\frac{1}{\sin A} + 1}}}$

$\\ \sf \implies {\sf \dfrac{ \frac{1 }{ \sin A} }{\frac{1 -\sin A }{\sin A} } + {\sf \dfrac{ \frac{1}{ \sin A} }{\frac{1 + \sin A}{\sin A} }}}$

$\\ \sf\implies \sf \dfrac{1 }{ \cancel{\sin A}} \times \dfrac{ \cancel{\sin A}}{1 -\sin A} + \dfrac{1 }{ \cancel{\sin A}} \times \dfrac{ \cancel{\sin A}}{1 + \sin A}$

$\\ \sf\implies \sf \dfrac{ 1}{1 -\sin A} + \dfrac{ 1}{1 + \sin A}$

$\\ \sf\implies \sf \dfrac{(1 + \sin A) + (1 -\sin A)}{(1 -\sin A)(1 + \sin A)}$

$\\ \sf\implies \sf \dfrac{1 + \sin A + 1 -\sin A}{1 -\sin^{2} A}$

$\\ \sf\implies \sf \dfrac{2 }{ \cos^{2} A}$

$\\ \sf\implies \boxed{ \sf 2 \sec^{2} {A} }$

• LHS = RHS

HENCE PROVED