how do u prove : sec(A-B) = cos(A+B) / ( cosA^2 - sinB^2 )
Use the subtraction and addition formulas.
rewrite the left side as:
$\displaystyle \frac{1}{cosAcosB+sinAsinB}$
Multiply top and bottom by $\displaystyle cosAcosB-sinAsinB$
and you should be able to hammer it into the right side.
$\displaystyle \frac{1}{cosAcosB+sinAsinB}\cdot\frac{cosAcosB-sinAsinB}{cosAcosB-sinAsinB}$
$\displaystyle \frac{cosAcosB-sinAsinB}{cos^{2}Acos^{2}B-sin^{2}A\cdot{sin^{2}}B}$
$\displaystyle \frac{\overbrace{cosAcosB-sinAsinB}^{\text{cos(A+B)}}}{cos^{2}A-sin^{2}B}$
$\displaystyle \frac{cos(A+B)}{cos^{2}A-sin^{2}B}$