a) $A=\left({\sin }^2{3}^{\circ }+{\sin }^{2}87^{\circ }\right)+\left({\sin }^{2}15^{\circ }+{\sin }^{2}75^{\circ }\right)$

\begin{aligned} &=\left(\sin ^{2} 3^{\circ}+\cos ^{2} 3^{\circ}\right)+\left(\sin ^{2} 15^{\circ}+\cos ^{2} 15^{\circ}\right) \\ &=1+1=2. \end{aligned}​=(sin23∘+cos23∘)+(sin215∘+cos215∘)=1+1=2.​

b) $B=\left(\cos 0^{\circ }+\cos 180^{\circ }\right)+\left(\cos 20^{\circ }+\cos 160^{\circ }\right)+\dots +\left(\cos 80^{\circ }+\cos 100^{\circ }\right)$

\begin{aligned} &=\left(\cos 0^{\circ}-\cos 0^{\circ}\right)+\left(\cos 20^{\circ}-\cos 20^{\circ}\right)+\ldots+\left(\cos 80^{\circ}-\cos 80^{\circ}\right) \\ &=0. \end{aligned}​=(cos0∘−cos0∘)+(cos20∘−cos20∘)+…+(cos80∘−cos80∘)=0.​

c)  $\begin{array}{rl}C& =\left(\tan 5^{\circ }\tan 85^{\circ }\right)\left(\tan 15^{\circ }\tan 75^{\circ }\right)\dots \left(\tan 45^{\circ }\tan 45^{\circ }\right)\\ & =\left(\tan 5^{\circ }\cot 5^{\circ }\right)\left(\tan 15^{\circ }\cot 5^{\circ }\right)\dots \left(\tan 45^{\circ }\cot 5^{\circ }\right)\\ & =1.\end{array}$

a) $A=a^2\cdot 1+b^2\cdot 0+c^2\cdot (-1)=a^2-c^2$

b) $B=3-(1{)}^2+2{\left(\genfrac{}{}{0ex}{}{1}{2}\right)}^2-3{\left(\genfrac{}{}{0ex}{}{\sqrt{2}}{2}\right)}^2=1$

c) $C={\sin }^{2}45^{\circ }+3{\cos }^{2}45^{\circ }-2\left({\sin }^{2}50^{\circ }+{\sin }^{2}40^{\circ }\right)+4\tan 55^{\circ }\cdot \cot 55^{\circ }$
$C={\left(\genfrac{}{}{0ex}{}{\sqrt{2}}{2}\right)}^2+3{\left(\genfrac{}{}{0ex}{}{\sqrt{2}}{2}\right)}^2-2\left({\sin }^{2}50^{\circ }+{\cos }^{2}40^{\circ }\right)+4=\genfrac{}{}{0ex}{}{1}{2}+\genfrac{}{}{0ex}{}{3}{2}-2+4=4$