27. Suppose $a$ and $b$ are the roots of $x^{2}+x \sin \alpha+1=0$ while $c$ and $d$ are the roots of the equation $x^{2}+x \cos \alpha-1=0$. Find the value of $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}$.
27. Answer: 1
We have $a b=1$. Hence $a^{2}=\frac{1}{b^{2}}$ and $b^{2}=\frac{1}{a^{2}}$.
Also, $c d=-1$. Hence $d^{2}=\frac{1}{c^{2}}$ and $c^{2}=\frac{1}{d^{2}}$.
Hence $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}=a^{2}+b^{2}+c^{2}+d^{2}$
$$
\begin{array}{l}
=(a+b)^{2}-2 a b+(c+d)^{2}-2 c d \\
=\sin ^{2} \alpha-2+\cos ^{2} \alpha+2 \\
=1
\end{array}
$$