a/
\(tana+tanb=\frac{sina}{cosa}+\frac{sinb}{cosb}=\frac{sinacosb+cosa.sinb}{cosa.cosb}=\frac{sin\left(a+b\right)}{cosa.cosb}\)
\(C=tan80\left(tan20+tan140\right)+tan20.tan120\)
\(C=tan80.\frac{sin160}{cos20.cos140}+\frac{sin20.sin140}{cos20.cos140}\)
\(C=\frac{sin80}{cos80}.\frac{2.sin80.cos80}{\frac{1}{2}\left(cos160+cos120\right)}+\frac{-\frac{1}{2}\left(cos160-cos120\right)}{\frac{1}{2}\left(cos160+cos120\right)}\)
\(C=\frac{4sin^280}{cos160+cos120}-\frac{cos160-cos120}{cos160+cos120}\)
\(C=\frac{2\left(1-cos160\right)-cos160+cos120}{cos160+cos120}=\frac{2+cos120-3cos160}{cos120+cos160}\)
\(C=\frac{2-\frac{1}{2}-3cos160}{-\frac{1}{2}+cos160}=\frac{3-6cos160}{2cos160-1}=-3\)
b/
\(cos^275-sin^275=cos150=-\frac{\sqrt{3}}{2}\)
