If tan⁡θ+sin⁡θ=m and tan⁡θ−sin⁡θ=n,  then

From the given relations, m+n=2tan⁡θ,m−n=2sin⁡θthus , m2−n2=4tan⁡θsin⁡θ-----ialso , mn=tan2⁡θ−sin2⁡θ=sin⁡θtan⁡θ----iiFrom Eqs. (i) and (ii), we get m2−n2=4mn