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If `15 sin^(4)alpha+10cos^(4)alpha=6`, then the value of `8 cosec^(6)alpha+27 sec^(6)alpha` isA. 150B. 175C. 225D. 250

Answer» Correct Answer - D
`15 sin^(4)alpha+10 cos^(4)alpha=6`
Dividing by `cos^(4)alpha`, we get
`15 tan^(4)alpha+10=6 sec^(4)alpha`
`rArr 15 tan^(4)alpha+10=6(1+tan^(2)alpha)^(2)`
`rArr 9tan^(4)alpha-12tan^(2)alpha+4=0`
`rArr (3 tan^(2)alpha-2)^(2)=0`
`rArr tan^(2)alpha=(2)/(3)`
Now `8 cosec^(6)alpha+27sec^(6)alpha`
`8(1+cot^(2)alpha)^(3)+27(1+tan^(2)alpha)^(3)`
`=8(1+(3)/(2))^(3)+27(1+(2)/(3))^(3)`
`=125+125=250`


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