If tan θ + cot θ = 4, then the ratio of $3(tan^2\, θ + cot^2\, θ) \, to \, (2\,  cosec^2\, θ\,  sec^2\,  θ - 4) $ will be :

3 : 2

tanθ + cotθ = 4

On squaring both side

(tanθ + cotθ)² = 4²

tan²θ + cot²θ + 2tanθ.cotθ = 16

tan²θ + cot²θ = 16- 2     { tanθ = \(\frac{1}{cotθ}\)

tan²θ + cot²θ = 14

Now,

$3(tan^2\, θ + cot^2\, θ) \, to \, (2\,  cosec^2\, θ\,  sec^2\,  θ - 4) $

3 (tan²θ + cot²θ ) : 2 ( 1 + cot²θ ) ( 1 + tan²θ ) - 4

3 (tan²θ + cot²θ ) : 2 ( 1 + cot²θ+ tan²θ + 2cot²θ tan²θ ) - 4

3 ( 14 )   :   2 × ( 1 + 14 + 1 ) - 4 

42 : 28

3   :  2