TAN }^6}{\rm{\theta }} - 3{{\sec }^2}\theta {{\tan }^2} + 1}}{{{{\COS }^4}\theta - {{\sin }^4}\theta + 2si{n^2}\theta + 2\;}}\)Put θ = 0⇒ (1 – 0 – 0 + 1)/(1 + 0 + 0 + 2)⇒ 2/3 (satisfied)Detailed Method:\(\frac{{{{\sec }^6}\theta - {{\tan }^6}{\rm{\theta }} - 3{{\sec }^2}\theta {{\tan }^2} + 1}}{{{{\cos }^4}\theta - {{\sin }^4}\theta + 2si{n^2}\theta + 2\;}}\)\( \Rightarrow \frac{{{{({{\sec }^2}\theta )}^3} - {{({{\tan }^2}{\rm{\theta }})}^2} - 3{{\sec }^2}\theta {{\tan }^2} + 1}}{{\left( {{{\cos }^2}{\rm{\theta }} + {{\sin }^2}{\rm{\theta }}} \right)({{\cos }^2}\theta - {{\sin }^2}\theta ) + 2si{n^2}\theta + 2\;}}\)\( \Rightarrow \frac{{({{\sec }^2}\theta - {\rm{TA}}{{\rm{n}}^2}\theta )\left( {{\rm{se}}{{\rm{c}}^4}\theta + {\rm{ta}}{{\rm{n}}^4}\theta + {\rm{se}}{{\rm{c}}^2}\theta {\rm{ta}}{{\rm{n}}^2}\theta } \right) - 3{{\sec }^2}\theta {{\tan }^2} + 1}}{{{{\cos }^2}\theta + {\rm{si}}{{\rm{n}}^2}\theta + \;2}}\)\( \Rightarrow \frac{{{{\sec }^4}\theta + {{\tan }^4}\theta + {{\sec }^2}\theta {{\tan }^2}\theta - 3{{\sec }^2}\theta {{\tan }^2} + 1}}{{1 + 2}}\)\(\Rightarrow \frac{{{{\sec }^4}\theta + {{\tan }^4}\theta - 2{{\sec }^2}\theta {{\tan }^2}\theta + 1}}{3}\)\( \Rightarrow \frac{{{{({{\sec }^2}\theta - {{\tan }^2}\theta )}^2} + 1}}{3}\)⇒ (1 + 1)/3⇒ 2/3