GIVEN:

\(\left[ {\frac{{\left( {sin\theta \; + \;cos\theta \; + \;1} \right)\left( {sin\theta \; + \;cos\theta - 1} \right)sec\theta \;cosec\theta }}{{\left( {co{s^6}\theta \; + \;si{n^6}\theta \; + \;3si{n^2}\theta \;co{s^2}\theta } \right)}}} \right]\)

FORMULA USED:

(a + b)(a- b) = a² – b²

\(co{s^6}\theta \; + \;si{n^6}\theta = 1 - 3{\sin ^2}\theta {\cos ^2}\theta \)

sin²θ + cos²θ = 1

CALCULATION:

\(\left[ {\frac{{\left( {sin\theta \; + \;cos\theta \; + \;1} \right)\left( {sin\theta \; + \;cos\theta - 1} \right)sec\theta \;cosec\theta }}{{\left( {co{s^6}\theta \; + \;si{n^6}\theta \; + \;3si{n^2}\theta \;co{s^2}\theta } \right)}}} \right]\)

⇒\(\left[ {\frac{{\left[ {{{\left( {sin\theta \; + \;cos\theta } \right)}^2} - 1} \right]sec\theta \;cosec\theta }}{{\left( {1 - 3{{\sin }^2}\theta {{\cos }^2}\theta \; + \;3si{n^2}\theta \;co{s^2}\theta } \right)}}} \right]\)

⇒\(\left[ {\frac{{({{\sin }^2}\theta \; + \;{{\cos }^2}\theta \; + \;2sin\theta \;cos\theta - 1)sec\theta \;cosec\theta }}{{\left( 1 \right)}}} \right]\)

⇒\(\left[ {\frac{{\left( {1\; + \;2sin\theta \;cos\theta - 1} \right)sec\theta \;cosec\theta }}{{\left( 1 \right)}}} \right]\)

⇒ 2sin θ cos θ sec θ cosec θ

⇒ 2