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

Options

A. 125
B. 250
C. 50
D. 75

Answer

B

Explanation

We have,
`10sin^(4)alpha+15 cos^(4)alpha=6`
`implies10 sin^(4)alpha+15cos^(4)alpha=6(sin^(2)alpha+cos^(2)alpha)^(2)`
`implies10 tan^(4)alpha+15=6(tan^(2)alpha+1)^(2)`
`implies(2 tan^(2)alpha-3)^(2)=0impliestan^(2)alpha=3/2`
`therefore25cosec^(6)alpha+8 sec^(6)alpha`
`=27(1+cot^(2)alpha)^(3)+8(1+tan^(2)alpha)^(3)`
`=27(1+(2)/(3))^(3)+8(1+(3)/(2))^(3)=250`