Explain the following:

Dear Student,

G i v e n △ = a^{2} - (b - c)^{2} N o w a s w e k n o w t h a t s = \frac{a + b + c}{2} \Rightarrow a = 2 s - (b + c) P u t t i n g t h i s v a l u e o f a i n △ = a^{2} - (b - c)^{2}, w e g e t △ = {\{2 s - (b + c)\}}^{2} - (b - c)^{2} \Rightarrow △ = \{4 s^{2} + (b + c)^{2} - 4 s (b + c)\} - (b - c)^{2} \Rightarrow △ = 4 s^{2} - 4 s (b + c) + \{(b + c)^{2} - (b - c)^{2}\} \Rightarrow △ = 4 s^{2} - 4 s (b + c) + 4 b c \Rightarrow △ = 4 s^{2} - 4 s b - 4 s c + 4 b c \Rightarrow △ = 4 s (s - b) - 4 c (s - b) \Rightarrow △ = (4 s - 4 c) (s - b) \Rightarrow △ = 4 (s - b) (s - c) \Rightarrow \frac{1}{4} = \frac{(s - b) (s - c)}{△} . . . . . . . . . . (1) N o w a s w e k n o w t h a t \tan \frac{A}{2} = \sqrt{\frac{(s - b) (s - c)}{s (s - a)}} \Rightarrow \sqrt{(s - b) (s - c)} = \tan \frac{A}{2} \sqrt{s (s - a)} M u l t i p l y i n g b o t h s i d e s b y \sqrt{(s - b) (s - c)} \Rightarrow (s - b) (s - c) = \tan \frac{A}{2} △ (∵ △ = \sqrt{s (s - a) (s - b) (s - c)}) \Rightarrow \frac{(s - b) (s - c)}{△} = \tan \frac{A}{2} . . . . . . . . . . (2) ∴ \tan \frac{A}{2} = \frac{1}{4} \{u \sin g (1) a n d (2)\} \Rightarrow \tan A = \frac{2 \tan \frac{A}{2}}{1 - \tan^{2} \frac{A}{2}} = \frac{2 \times \frac{1}{4}}{1 - (\frac{1}{4})^{2}} = \frac{\frac{1}{2}}{\frac{15}{16}} = \frac{8}{15}

Thus Correct Option (B)

Regards