Answer:
Option D
Explanation:
We have,
c^{2}x^{2}-c(a+b)x+ab=0
\Rightarrow c^{2}x^{2}-cax-cbx+ab=0
\Rightarrow cx(cx-a)-b(cx-a)=0
\Rightarrow (cx-a)(cx-b)=0
\Rightarrow cx-a=0 or cx-b=0
\Rightarrow x=\frac{a}{c} and x=\frac{b}{c}
As, sin A and sin B are roots of the given equation
Let \sin A= \frac{a}{c} and \sin B =\frac{b}{c}
\Rightarrow c= \frac{a}{\sin A} and c=\frac{b}{\sin B}
\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=c ...........(i)
Using sine law, \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} .............(ii)
So, \frac{c}{\sin c}=c [From eqs.(i) and (ii) ]
\Rightarrow \sin c= \frac{c}{c}
\Rightarrow \ sin C=1 [\because \sin \frac{\pi}{2}=1]
\Rightarrow \angle C=\frac{\pi}{2}

\sin A= \frac{a}{c} and \cos A= \frac{b}{c}
\therefore \sin A+\cos A= \frac{a}{c}+\frac{b}{c}= \frac{a+b}{c}