a/ \(\frac{A}{2}+\left(\frac{B}{2}+\frac{C}{2}\right)=90^0\)
\(\Rightarrow sin\frac{A}{2}=cos\left(\frac{B}{2}+\frac{C}{2}\right)=cos\frac{B}{2}cos\frac{C}{2}-sin\frac{B}{2}.sin\frac{C}{2}\)
b/ \(\frac{tan^2A-tan^2B}{1-tan^2A.tan^2B}=\frac{\left(tanA-tanB\right)}{\left(1+tanA.tanB\right)}.\frac{\left(tanA+tanB\right)}{\left(1-tanA.tanB\right)}=tan\left(A-B\right).tan\left(A+B\right)\)
\(=tan\left(A-B\right).tan\left(180^0-C\right)=-tan\left(A-B\right).tanC\)
c/
\(A+B+C=180^0\Rightarrow cot\left(A+B\right)=-cotC\)
\(\Leftrightarrow\frac{cotA.cotB-1}{cotA+cotB}=-cotC\)
\(\Leftrightarrow cotA.cotB-1=-cotA.cotC-cotB.cotC\)
\(\Leftrightarrow cotA.cotB+cotB.cotC+cotA.cotC=1\)
