Answer
See the steps.
Work Step by Step
$LHS = \sec{(A+B)} = \dfrac{1}{\cos{(A+B)}} = \dfrac{1}{\cos{A} \cos{B} - \sin{A} \sin{B}}$
Multiplying by $\dfrac{\cos{A} \cos{B} + \sin{A} \sin{B}}{\cos{A} \cos{B} + \sin{A} \sin{B}}$
$LHS = \dfrac{\cos{A} \cos{B} + \sin{A} \sin{B}}{(\cos{A} \cos{B} - \sin{A} \sin{B}) (\cos{A} \cos{B} + \sin{A} \sin{B})}$
$LHS = \dfrac{\cos{(A-B)}}{\cos^2{A} \cos^2 {B} - \sin^2 {A} \sin^2 {B}}$
Denominator $ = \cos^2{A}(1-\sin^2{B}) - (1- \cos^2 {A}) \sin^2 {B}$
Denominator $ = \cos^2 {A} - \cos^2 {A} \sin^2 {B} - \sin^2 {B} + \cos^2 {A} \sin^2 {B}$
Denominator $ = \cos^2{A} - \sin^2{B}$
$LHS = \dfrac{\cos{(A-B)}}{\cos^2{A} - \sin^2{B}} = RHS$
