If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$?
$\frac{1}{2}$
$\frac{1}{2}$
$1$
$2$
If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 \beta)$?
$\frac{13}{75}$
$\frac{13}{85}$
$\frac{17}{85}$
$\frac{19}{85}$
What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$?
$0$
$1$
$2$
$4$
Consider the following statements:
1. In a triangle $ABC$, if $\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is an acute-angled triangle.
2. In a triangle $ABC$, if $\tan A \cdot \tan B \cdot \tan C > 0$, then the triangle is an obtuse-angled triangle.
Which of the statements given above is/are correct?
1 only
2 only
Both 1 and 2
Neither 1 nor 2