If `sin^ 4 x/2+cos^4 x/3 =1/5` then

Options

A. `tan^(2)x=2/3`
B. `(sin^(8)x)/(8)+(cos^(8)x)/(27)=1/125`
C. `tan^(2)x=1/3`
D. `(sin^(8)x)/(8)+(cos^(8)x)/(27)=2/125`

Answer

A, B

Explanation

We have,
`(sin^(4)x)/(2)+(cos^(4)x)/(3)=1/5`
`implies((2+3))/(2)sin^(4)x+((2+3))/(3)cos^(4)x=1`
`implies(1+(3)/(2))sin^(4)x+((2)/(3)+1)cos^(4)x=(sin^(2)x+cos^(2)x)^(2)`
`implies(sqrt((3)/(2))sin^(2)x)^(2)+(sqrt((2)/(3))cos^(2)x)^(2)-2sinx cos^(2)x=0`
`implies(sqrt((3)/(2))sin^(2)x=sqrt((2)/(3))cos^(2)x)=0`
`impliessqrt((3)/(2))sin^(2)x=sqrt((2)/(3))cos^(2)x`
`impliestan^(2)x=2/3and (sin^(2)x)/(2)=(cos^(2)x)/(3)`
Now, `(sin^(2)x)/(2)=(cos^(2)x)/(3)`
`implies(sin^(2)x)/(2)=(cos^(2)x)/(3)-(sin^(2)x+cos^(2)x)/(2+3)`
`impliessin^(2)x=2/5and cos^(2)x=3/5`
`(sin^(2)x)^(4)=16/625and (cos^(2)x)^(4)=81/625`
`implies(sin^(8)x)/(8)+(cos^(8)x)/(27)=(2)/(625)+(3)/(625)=(1)/(125)`