If 10 sin^4 θ + 15 cos^4 θ = 6, then find the value of 27 cosec^6 θ + 8 sec^6 θ/16 sec^8 θ

Question

If \(10 \sin^4 \theta + 15 \cos^4 \theta = 6, \) then find the value of \(\frac{27\ cosec^6 \theta \ +\ 8 \sec^6 \theta}{16 \sec^8 \theta}\) 

Answer 1

Answer is: \(\frac{2}{5}\) 

Let \(\sin^2 \theta = t\)  

\(10t^2 + 15(1 + t^2 – 2t) = 6\)  

\(25t^2 – 30t + 9 = 0\)   

\((5t – 3)^2 = 0\)