Trigonometrical Equations question bank (Mathematics)

In $\triangle ABC$,with usual notations,$m \angle C = \frac{\pi}{2}$. If $\tan \left(\frac{A}{2}\right)$ and $\tan \left(\frac{B}{2}\right)$ are the roots of the equation $a_1 x^2 + b_1 x + c_1 = 0$ $(a_1 \neq 0)$,then:

If the sides of a triangle $a, b, c$ are in $A.P.$,then with usual notations,$a \cos ^2 \frac{C}{2} + c \cos ^2 \frac{A}{2}$ is