if `0<=x<=pi` and `81^(sin^2x)+81^(cos^2x)=30` then `x=`

Options

A. `(pi)/(6)`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `(3pi)/(4)`

Answer

A

Explanation

Given `81^(sin^(2)x)+81^(cos^(2)x)=30`
`implies81^(sin^(2)x)+81^(1-sin^(2)x)=30`
`implies81^(sin^(2)x)+(81)/(81^(sin^(2)x))=30`
`y+(81)/(y)=30" "["put"81^(sin^(2)x)=y]`
`impliesy^(2)-30y+81=0`
`implies(y-27)(y-3)=0`
`implies81^(sin^(2)x)=27or81^(sin^(2)x)=3`
`implies3^(4sin^(2)x)=3^(3)or81^(sin^(2)x)=3`
`implies3^(4sin^(2)x)=3^(3)or81^(sin^(2)x)=3`
`impliessinx=(sqrt3)\/(2)orsinx=(1)/(2)`
`impliesx=(pi)/(3),(2pi)/(3)orx=(pi)/(6),(5pi)/(6)`