[latex]a) (cosx+tgxsinx)ctgx=frac{1}{sinx}\ L=(cosx+frac{sinx}{cosx}cdot sinx)cdotfrac{cosx}{sinx}=(cosx+frac{sin^2x}{cosx})cdotfrac{cosx}{sinx}=\ =(frac{cos^2x}{cosx}+frac{sin^2x}{cosx})cdotfrac{cosx}{sinx}=frac{1}{cosx}cdotfrac{cosx}{sinx}=frac{1}{sinx}\ \ L=P[/latex] [latex]b) frac{1}{1+cosx}=frac{1}{sin^2x}-frac{1}{sinx}ctgx\ \ P=frac{1}{sin^2x}-frac{1}{sinx}ctgx=frac{1}{1-cos^2x}-frac{1}{sinx}cdotfrac{cosx}{sinx}=\ =frac{1}{1-cos^2x}-frac{cosx}{sin^2x}=frac{1}{1-cos^2x}-frac{cosx}{1-cos^2x}=\ =frac{1-cosx}{1-cos^2x}=frac{1-cosx}{(1-cosx)(1+cosx)}=frac{1}{1+cosx}\ \ L=P[/latex]
