7. Suppose that $(1+\sec \theta)(1+\csc \theta)=6$. Determine the value of $(1+\tan \theta)(1+\cot \theta)$.
Answer: $\frac{49}{12}$
Solution: The equation is equivalent to $1+\sin \theta+\cos \theta=5 \sin \theta \cos \theta$. Let $A:=\sin \theta+\cos \theta$ and $B:=\sin \theta \cos \theta$. Then $1+A=5 B$ and $A^{2}=1+2 B$. As $1+A \neq 0$,
$$
1+A=\frac{5\left(A^{2}-1\right)}{2} \Longrightarrow 1=\frac{5(A-1)}{2},
$$
which gives $A=\frac{7}{5}$. Consequently, we get $B=\frac{12}{25}$. Hence,
$$
(1+\tan \theta)(1+\cot \theta)=\frac{A^{2}}{B}=\frac{49}{25} \cdot \frac{25}{12}=\frac{49}{12} .
$$