If `sintheta + 2 costheta=1`,then `2sintheta-costheta=`

Options

A. `1`
B. `3`
C. `0`
D. `2`

Answer

D

Explanation

Given, `sintheta+2costheta=1`
On squaring both sides, we get
`(sintheta+2costheta)^(2)=1`
`sin^(2)theta+4cos^(2)theta+4sintheta.costheta=1`
`rArr (1-cos^(2)theta) + 4(1-sin^(2)theta)+4sintheta.costheta=1` `[therefore sin^(2)theta+ cos^(2)theta=1]`
`rArr 4sin^(2)theta + cos^(2)theta-4sintheta. costheta=4`
`rArr (2sintheta-costheta)^(2)=4` `[therefore a^(2)+b^(2)-2ab=(a-b)^(2)]`
`rArr 2sintheta-costheta=2` Hence proved.