If `cosectheta - sintheta = m` and `sectheta -costheta,= n` , eliminate `theta`

Explanation

Given,
`cosectheta-sintheta=m`
or `1/sintheta-sintheta=m`
or `(1-sin^2theta)/sintheta=m" or " (cos^2theta)/sintheta=m (i)`
Also `sectheta-costheta=n`
or `1/costheta-costheta=n`
`or (1-cos^2theta)/(costheta)=n" or "(sin^2theta)/costheta=n (ii)`
From Eq.(i), `sintheta=(cos^2theta)/m (iii)`
Putting the value of `sintheta `in Eq.(ii), we get
`(cos^4theta)(m^2costheta)=n" or " cos^3theta=m^2n`
`:. costheta=(m^2n)^(1//3)" or " cos^2theta=(m^2n)^(2//3) (iv)`
From Eq. (iii), `sintheta=(cos^2theta)/m=((m^2n)(2//3))/m`
`=(m^(4//3)n^(2//3))/m=m^(1//3)n^(2//3)=(mn^2)^(1//3)`
`:. sin^2theta=(mn^2)^(2//3) (v)`
Adding Eqs. (iv) and (v), we get
`(m^2n)^(2//3)+(mn^2)^(2//3)=cos^2theta+sin^2theta`
or `(m^2n)^(2//3)=1`