Geometry
# Sum and Difference Trigonometric Formulas

$\large \dfrac{\cot^375^{\circ}+\tan^375^{\circ}}{\cot75^{\circ}+\tan75^{\circ}} = \ ?$

$\large \dfrac{\sin^2 (3A)}{\sin^2 (A)}-\dfrac{\cos^2 (3A)}{\cos^2 (A)}=2$

$\implies \large \cos (2A)= \ ?$

$\large \tan(63^\circ) = \sqrt{\sqrt a-\sqrt b} + \sqrt{\sqrt c-\sqrt b}$

The equation above is true for positive integers $a,b,$ and $c.$ What is the value of $a+b+c?$