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Answer 1

$1)\; \; tga+\frac{cosa}{1-sina}=tga+\frac{cos^2\frac{a}{2}-sin^2\frac{a}{2}}{sin^2\frac{a}{2}+cos^2\frac{a}{2}-2sin\frac{a}{2}\, cos\frac{a}{2}}=\\\\=tga+\frac{(cos\frac{a}{2}-sin\frac{a}{2})(cos\frac{a}{2}+sin\frac{a}{2})}{(cos\frac{a}{2}-sin\frac{a}{2})^2}=tga+\frac{cos\frac{a}{2}+sin\frac{a}{2}}{cos\frac{a}{2}-sin\frac{a}{2}}=tga+\frac{1+tg\frac{a}{2}}{1-tg\frac{a}{2}}=\\\\=tga+\frac{tg\frac{\pi}{4}+tg\frac{a}{2}}{1-tg\frac{\pi}{4}\cdot tg\frac{a}{2}}=tga+tg(\frac{a}{2}+\frac{\pi}{4})$

$2)\; \; ctga+\frac{sina}{1+cosa}=ctga+\frac{2sin\frac{a}{2}\cdot cos\frac{a}{2}}{(sin^2\frac{a}{2}+cos^2\frac{a}{2})+(cos^2\frac{a}{2}-sin^2\frac{a}{2})}=\\\\=ctga+\frac{2sin\frac{a}{2}\cdot cos\frac{a}{2}}{2cos^2\frac{a}{2}}=ctga+\frac{sin\frac{a}{2}}{cos\frac{a}{2}}=ctga+tg\frac{a}{2}}=\\\\\star \; =\frac{cosa}{sina}+\frac{sin\frac{a}{2}}{cos\frac{a}{2}}=\frac{cosa\cdot cos\frac{a}{2}+sina\cdot sin\frac{a}{2}}{sina\cdot cos\frac{a}{2}}=\frac{cos(a-\frac{a}{2})}{sina\cdot cos\frac{a}{2}}=\frac{cos\frac{a}{2}}{sina\cdot cos\frac{a}{2}}=\frac{1}{sina}$

$P.S.\; \; \; \frac{cos\frac{a}{2}+sin\frac{a}{2}}{cos\frac{a}{2}-sin\frac{a}{2}}=\Big [\; \frac{:cos\frac{a}{2}}{:cos\frac{a}{2}}\Big ]=\frac{1+tg\frac{a}{2}}{1-tg\frac{a}{2}}=\frac{1+tg\frac{a}{2}}{1-1\cdot tg\frac{a}{2}}=\Big [\; 1=tg\frac{\pi}{4}\; \Big ]=\\\\=\frac{tg\frac{\pi}{4}+tg\frac{a}{2}}{1-tg\frac{\pi}{4}\cdot tg\frac{a}{2}}=tg(\frac{a}{2}+\frac{\pi}{4})\; ;$