If `0ltthetalt 90^(@)+sintheta=m and tan_(theta)-sin_(theta)=n`where m=b then the value of `m^(2)-n^(2)` is

Options

A. `2(tan^(2)theta+sin^(2)theta)`
B. 4mm
C. `4sqrtmm`
D. `2(m^(2)+n^(2))`

Answer

c

Explanation

`tan theta+sintheta= m`
squaring both sides
`tan^(2)theta+sin^(2)theta+2 tan thetasintheta=m^(2)`
`tan theta-sin theta=n`
squaring both sides
`tan^(2)theta+sin^(2)theta-2tan theta sin theta=n^(2)`
Subtract from (i) `&` (ii)
`m^(2)-n^(2)=tan^(2)theta+sin^(2)+2tan theta`
`sin theta-tan^(2)theta-sin^(2)theta+2tan theta`
`sin theta`
`m^(2)-n^(2)=4tan theta sin theta`
` 4sqrt(tan^(2)thetasin^(2)theta)`
`=4sqrt(tan^(2)theta(1-cos^(1)theta))`
`=4sqrt(tan^(2)theta-sin^(2)theta)`
`=4sqrt(mm)`