You should remember that `tan A = sinA/cosA` and `tan B = sinB/cos B` , hence, substituting `m sin B` for `sin A ` in `tan A = n tan B` yields:

`tan A = n tan B =>sinA/cosA = n(sinB/cos B) => (m sin B)/cosA = n (sinB/cos B)`

Reducing by duplicate factors yields:

`m/cos A = n/cos B => (m/cos A)^2 = (n/cos B)^2`

`(cos A)^2 = (m/n)^2*cos^2 B`

Substituting `1 - sin^2B ` for `cos^2 B` yields:

`(cos A)^2 = (m/n)^2*(1 - sin^2B)`

Since `sin^2B = (sin^2A)/m^2` yields:

`(cos A)^2 = (m/n)^2*(1 - (sin^2A)/m^2)`

**Hence, evaluating `(cos A)^2` under the given conditions yields `(cos A)^2 = (m/n)^2*(1 - (sin^2A)/m^2).` **