Shortcut Trick Given equation is tan2θ+3secθ−9=0 How to think about trick: Since R.H.S is a rational number So, L.H.S should be rational for this 3secθ must be rational because tan2θ and 9 are rational number Now, we have to think that on which θ,secθ is rational so we get θ=60∘ We can't take 0∘ or 90∘ according to the question Process: Put θ=60∘ in the given equation We get, LHS =tan260∘+3sec60∘−9 ⇒3+3×2−9=0= RHS Now, We have to find the value of 12cot2θ+3cosecθ, at θ=60∘ ⇒12cot260∘+cosec60∘ ⇒12(1∕3)+3×2∕√3 ⇒4+2√3=(√3)2+(1)2+2⋅(√3).(1) Using formula a2+b2+2ab=(a+b)2 ⇒(√3+1)2 ∴ The required value is (√3+1)2