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

Question

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

(1) \(\frac{3}{5}\)

(2) \(\frac{2}{5}\)

(3) \(\frac{1}{5}\)

(4) \(\frac{4}{5}\)

Answer 1

Correct option is: (1) \(\frac{3}{5}\) 

\(10\left(\sin ^{4} \theta+\cos ^{4} \theta\right)+5 \cos ^{4} \theta=6\)

\(\Rightarrow 10\left[1-2 \sin ^{2} \theta \cos ^{2} \theta\right]+5 \cos ^{4} \theta=6 \)

\(\Rightarrow 25 \cos ^{4} \theta-20 \cos ^{2} \theta+4=0 \)

\(\Rightarrow \left(5 \cos ^{2} \theta-2\right)^{2}=0 \)