The sum of all values of `theta in (0,(pi)/(2))` satisfying `sin^(2)2theta+cos^(4)2theta=(3)/(4)` is :

Options

A. `pi`
B. `(3pi)/8`
C. `(5pi)/4`
D. `(pi)/2`

Answer

D

Explanation

`sin^(2)2theta+cos^(4)2theta=3/4`
Put `sin^(2)2theta=t`
`t+(1-t)^(2)=3/4`
`t+t^(2)+1-2t=3/4`
`t^(2)-t+1/4=0`
`(t-1/2)^(2)=0`
`t=1/2`
`sin^(2)2theta=1/2`
`sin^(2)2theta=pm1/(sqrt2)`
`2theta=(pi)/4,(3pi)/4`
`theta=(pi)/8,(3pi)/8`
`(pi)/8+(3pi)/8=(pi)/2`