What is $\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$ equal to?
$-\frac{\pi}{4}$
$\frac{\pi}{4}$
$\tan^{-1} \left( \frac{a^2 - b^2}{a^2 + b^2} \right)$
$\tan^{-1} \left( \frac{2ab}{a^2 + b^2} \right)$
Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta \in [0, \pi]$?
$1-\cos \beta <0$
$1-\cos \beta \leq 0$
$1-\cos \beta>0$
$1-\cos \beta \geq 0$
In a triangle $ABC$, $AB=16 \text{ cm}, BC=63 \text{ cm}$ and $AC=65 \text{ cm}$. What is the value of $\cos 2A+\cos 2B+\cos 2C$?
$-1$
$0$
$1$
$\frac{76}{65}$
If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$?
$\frac{1}{2}$
$\frac{1}{2}$
$1$
$2$