We know that $ cosh^2x - sinh^2x = 1 \\ \\ \therefore 1 - cosh^2x = -sinh^2x and 1+sinh^2x = cosh^2x \; \; \; \ldots (i) $

$ \therefore L.H.S. = \tfrac{1}{ 1 - \tfrac{1}{1 - \tfrac{1}{cosh^2 x}} } = \tfrac{1}{ 1 - \tfrac{1}{1 + \tfrac{1}{sinh^2 x}} } \; \; \; \ldots From (i) $

$= \tfrac{1}{ 1 - \tfrac{1}{ \tfrac{1 + sinh^2x }{sinh^2 x}} } $

$ \tfrac{1}{ 1 - \tfrac{sinh^2x}{ cosh^2x} } \; \; \; \ldots From (i) $

$ \tfrac{1}{ \tfrac{cosh^2x - sinh^2x}{ cosh^2x} } = \tfrac{cosh^2x}{ {cosh^2x - sinh^2x} } = \tfrac{cosh^2x}{ 1 } = cosh^2x \; \; \; \; \; \ldots From \; \; (i) $

= R.H.S.