ln[(1+sinx)/(1-sinx)]^1/2=(1/2)ln(1+sinx)-(1/2)ln(1-sinx).
The derivative of ln(1+sinx) is (1/(1+sinx))(cosx)=cosx/(1+sinx). Similarly, the derivative of ln(1-sinx)=-cosx/(1-sinx). The whole derivative is (1/2)cosx(1/(1+sinx)+1/(1-sinx))=((cosx)/2)(1-sinx+1+sinx)/(1-sin^2x))=cosx/cos^2x=secx.