I3.1 Suppose that $a=\cos ^{4} \theta-\sin ^{4} \theta-2 \cos ^{2} \theta$, find the value of $a$.
I3.2 If $x^{y}=3$ and $b=x^{3 y}+10 a$, find the value of $b$.
I3.3 If there is (are) $c$ positive integer(s) $n$ such that $\frac{n+b}{n-7}$ is also a positive integer, find the value of $c$.

I3.4 Suppose that $d=\log _{4} 2+\log _{4} 4+\log _{4} 8+\ldots+\log _{4} 2^{c}$, find the value of $d$.
$\begin{array}{l} a=\left(\cos ^{2} \theta+\sin ^{2} \theta\right)\left(\cos ^{2} \theta-\sin ^{2} \theta\right)-2 \cos ^{2} \theta \\ =\cos ^{2} \theta-\sin ^{2} \theta-2 \cos ^{2} \theta=-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)=-1 \\ b=\left(x^{y}\right)^{3}-10=3^{3}-10=27-10=17 \\ \frac{n+17}{n-7}=1+\frac{24}{n-7} \\ n-7=1,2,3,4,6,8,12,24, c=8 \\ d=\log _{4} 2+\log _{4} 4+\log _{4} 8+\ldots+\log _{4} 2^{8} \\ =\log _{4}\left(2 \times 4 \times 8 \times \ldots \times 2^{8}\right)=\log _{4}\left(2^{1+2+3+\ldots+8}\right) \\ =\log _{4}\left(2^{36}\right)=\frac{\log 2^{36}}{\log 4}=\frac{36 \log 2}{2 \log 2}=18\end{array}$