(1÷sec^2theta-cos^2theta+1÷cosec^2theta-sin^2theta) sin^2theta cis^2theta =1-sin^2thetacos^2theta÷2+sin^2thetacos^2theta

LHS{tex} = \left( {\frac{1}{{{{\sec }^2}\theta - {{\cos }^2}\theta }} + \frac{1}{{\cos e{c^2}\theta - {{\sin }^2}\theta }}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{1}{{\frac{1}{{{{\cos }^2}\theta }} - {{\cos }^2}\theta }} + \frac{1}{{\frac{1}{{{{\sin }^2}\theta }} - {{\sin }^2}\theta }}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{1}{{\frac{{1 - {{\cos }^4}\theta }}{{{{\cos }^2}\theta }}}} + \frac{1}{{\frac{{1 - {{\sin }^4}\theta }}{{{{\sin }^2}\theta }}}}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{{{{\cos }^2}\theta }}{{1 - {{\cos }^4}\theta }} + \frac{{{{\sin }^2}\theta }}{{1 - {{\sin }^4}\theta }}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{{{{\cos }^2}\theta \left( {1 - {{\sin }^4}\theta } \right) + {{\sin }^2}\theta \left( {1 - {{\cos }^4}\theta } \right)}}{{\left( {1 - {{\cos }^4}\theta } \right)\left( {1 - {{\sin }^4}\theta } \right)}}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex}=\left( {\frac{{{{\cos }^2}\theta \left( {1 - {{\sin }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right) + {{\sin }^2}\theta \left( {1 - {{\cos }^2}\theta } \right)\left( {1 + {{\cos }^2}\theta } \right)}}{{\left( {1 - {{\cos }^2}\theta } \right)\left( {1 + {{\cos }^2}\theta } \right)\left( {1 - {{\sin }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{{{{\cos }^2}\theta \cdot {{\cos }^2}\theta \left( {1 + {{\sin }^2}\theta } \right) + {{\sin }^2}\theta {{\sin }^2}\theta \left( {1 + {{\cos }^2}\theta } \right)}}{{{{\sin }^2}\theta \left( {1 + {{\cos }^2}\theta } \right){{\cos }^2}\theta \left( {1 + {{\sin }^2}\theta } \right)}}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex} {tex}\left[ {\because {{\sin }^2}\theta + {{\cos }^2}\theta = 1} \right]{/tex}
{tex} = \left( {\frac{{{{\cos }^4}\theta \left( {1 + {{\sin }^2}\theta } \right) + {{\sin }^4}\theta \left( {1 + {{\cos }^2}\theta } \right)}}{{{{\sin }^2}\theta {{\cos }^2}\theta \left( {1 + {{\cos }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}} \right){\sin ^2}\theta {\cos ^2}\theta {/tex}
{tex} = \left( {\frac{{{{\cos }^4}\theta \left( {1 + {{\sin }^2}\theta } \right) + {{\sin }^4}\theta \left( {1 + {{\cos }^2}\theta } \right)}}{{\left( {1 + {{\cos }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}} \right){/tex}
{tex} = \left( {\frac{{{{\cos }^4}\theta + {{\cos }^4}\theta {{\sin }^2}\theta + {{\sin }^4}\theta + {{\sin }^4}\theta {{\cos }^2}\theta }}{{\left( {1 + {{\cos }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}} \right){/tex}
{tex} = \left( {\frac{{{{\cos }^4}\theta + {{\sin }^4}\theta + {{\cos }^4}\theta {{\sin }^2}\theta + {{\sin }^4}\theta {{\cos }^2}\theta }}{{\left( {1 + {{\cos }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}} \right){/tex}
{tex} = \frac{{{{\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)}^2} - 2{{\cos }^2}\theta {{\sin }^2}\theta + {{\cos }^2}\theta {{\sin }^2}\theta \left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)}}{{\left( {1 + {{\cos }^2}\theta } \right)\left( {1 + {{\sin }^2}\theta } \right)}}{/tex}
{tex} = \frac{{1 - 2{{\cos }^2}\theta {{\sin }^2}\theta + {{\cos }^2}\theta {{\sin }^2}\theta \times 1}}{{1 + {{\sin }^2}\theta + {{\cos }^2}\theta + {{\cos }^2}\theta {{\sin }^2}\theta }}{/tex} {tex}\left[ {{{\sin }^2}\theta + {{\cos }^2}\theta = 1} \right]{/tex}
{tex} = \frac{{1 - {{\cos }^2}\theta {{\sin }^2}\theta }}{{1 + 1 + {{\cos }^2}\theta {{\sin }^2}\theta }}{/tex}
{tex} = \frac{{1 - {{\cos }^2}\theta {{\sin }^2}\theta }}{{2 + {{\cos }^2}\theta {{\sin }^2}\theta }}{/tex}
= RHS
Hence proved