Explanation

(a)
$\sin{t + \cos{t = \frac{1}{5}}}$
$\Rightarrow \ \ \frac{2\tan\frac{t}{2}}{1 + \tan^{2}\frac{t}{2}} + \frac{1 - \tan^{2}\frac{t}{2}}{1 + \tan^{2}\frac{t}{2}} = \frac{1}{5}$
$\Rightarrow \ 10\tan{\frac{t}{2} + 5 - 5\tan^{2}{\frac{t}{2} = 1 + \tan^{2}\frac{t}{2}}}$
$\Rightarrow \ 6\tan^{2}{\frac{t}{2} - 10\tan{\frac{t}{2} - 4 = 0}}$
$\Rightarrow \ \ \left( 6\tan{\frac{t}{2} + 2} \right)\left( \tan{\frac{t}{2} - 2} \right) = 0$
$\Rightarrow \ \tan{\frac{t}{2} = \frac{- 1}{3},\ 2}\$for , $0 < t < \pi$
$\tan{\frac{t}{2} = 2}$