Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$, Then $$\alpha < \beta$$