Ещё вверху под цифрой 3) и номера 466. 467

[latex]1)\quad \frac{1-2cos\frac{x}{2}+cosx}{1+2cos\frac{x}{2}+cosx}=\frac{1-2cos\frac{x}{2}+(cos^2\frac{x}{2}-sin^2\frac{x}{2})}{1+2cos\frac{x}{2}+(cos^2\frac{x}{2}-sin^2\frac{x}{2})} =\\\\=\frac{(1-sin^2\frac{x}{2})-2cos\frac{x}{2}+cos^2\frac{x}{2}}{(1-sin^2\frac{x}{2})+2cos\frac{x}{2}+cos^2\frac{x}{2}} = \frac{cos^2\frac{x}{2}-2cos\frac{x}{2}+cos^2\frac{x}{2}}{cos^2\frac{x}{2}+2cos\frac{x}{2}+cos^2\frac{x}{2}} =[/latex] [latex]= \frac{2cos^2\frac{x}{2}-2cos\frac{x}{2}}{2cos^2\frac{x}{2}+2cos\frac{x}{2}}=\frac{2cos\frac{x}{2}(cos\frac{x}{2}-1)}{2cos\frac{x}{2}(cos\frac{x}{2}+1)}=\frac{-2sin^2\frac{x}{4}}{2cos^2\frac{x}{4}} =-ctg^2\frac{x}{4}\\\\2)\quad sinx=\frac{2tg\frac{x}{2}}{1+tg^2\frac{x}{2}}\; ,\; \; cosx=\frac{1-tg^2\frac{x}{2}}{1+tg^2\frac{x}{2}}\; ,\; \; tgx=\frac{2tg\frac{x}{2}}{1-tg^2\frac{x}{2}}\; ,\; ctgx=\frac{1-tg^2\frac{x}{2}}{2tg\frac{x}{2}}[/latex] [latex]sinx=2\cdot sin\frac{x}{2}\cdot cos\frac{x}{2}= \frac{2\cdot sin \frac{x}{2}\cdot cos \frac{x}{2} }{1} = \frac{2\cdot sin \frac{x}{2}\cdot cos \frac{x}{2}}{sin^2\frac{x}{2}+cos^2\frac{x}{2}} =[ \frac{:cos^2\frac{x}{2}}{:cos^2\frac{x}{2}}]=\\\\= \frac{2\cdot tg\frac{x}{2}}{1+tg^2\frac{x}{2}} [/latex] [latex]cosx=cos^2\frac{x}{2}-sin^2\frac{x}{2}= \frac{cos^2\frac{x}{2}-sin^2\frac{x}{2}}{1}= \frac{cos^2\frac{x}{2}-sin^2\frac{x}{2}}{cos^2\frac{x}{2}+sin^2\frac{x}{2}} =[ \frac{:cos^2\frac{x}{2}}{:cos^2\frac{x}{2}} ]=\\\\= \frac{1-tg^2\frac{x}{2}}{1+tg^2\frac{x}{2}}\\\\tgx=\frac{sinx}{cosx} = \frac{ \frac{2tg\frac{x}{2}}{1+tg^2\frac{x}{2}}}{\frac{1-tg^2\frac{x}{2}}{1+tg^2\frac{x}{2}}}= \frac{2tg\frac{x}{2}}{1-tg^2\frac{x}{2}} \; ,\; \; ctgx=\frac{1}{tgx}= \frac{1-tg^2\frac{x}{2}}{2tg\frac{x}{2}} [/latex] [latex]3)\quad t=tg\frac{a}{2}=3[/latex] [latex]\frac{2sina-3cosa}{4sina+5cosa}=\frac{2\cdot \frac{2t}{1+t^2}-3\cdot \frac{1-t^2}{1+t^2}}{4\cdot \frac{2t}{1+t^2}+5\cdot \frac{1-t^2}{1+t^2}}=\frac{4t-3+3t^2}{8t+5-5t^2}=\frac{12-3+27}{24+5-45}=\\\\=\frac{36}{-16}=-\frac{9}{4}[/latex]