$$RHS=\frac{1}{\sec A+\tan A}=\frac{1}{\frac{1+\sin A}{\cos A}}=\frac{\cos A}{1+\sin A}$$
Multiplying numerator and denominator by $(1-\sin A)$: $$=\frac{\cos A(1-\sin A)}{(1+\sin A)(1-\sin A)}=\frac{\cos A(1-\sin A)}{\cos^2A}=\frac{1-\sin A}{\cos A}$$
$$LHS=\sqrt{\frac{1-\sin A}{1+\sin A}}=\sqrt{\frac{(1-\sin A)^2}{(1+\sin A)(1-\sin A)}}=\sqrt{\frac{(1-\sin A)^2}{\cos^2A}}=\frac{1-\sin A}{\cos A}$$ Hence LHS = RHS.