[tex] \dfrac{\sin x+1}{\cos x}= \hbox{tg} \left( \dfrac{x}{2}+\dfrac{\pi}{4}ight)[/tex]

(sinx/2+cosx/2)^2/(cos^2x/2-sin^2x/2)=(sinx/2+cosx/2)/(cosx/2-sinx/2)=

=(tgx/2+1)/(1-tgx/2)=(tgП/4+tgx/2)/(1-tgП/4*tgx/2)=tg(x/2+П/4)