$\mid \begin{array}{ccc}1& 1& 1\\ 1+\mathrm{sinA}& 1+\mathrm{sinB}& 1+\mathrm{sinC}\\ \mathrm{sinA}+{\sin }^2\mathrm{A}& \mathrm{sinB}+{\sin }^2\mathrm{B}& \mathrm{sinC}+{\sin }^2\mathrm{C}\end{array}\mid =0$

⇒ sinBsin²C – sinCsin²B + sincsin²A – sinAsin²C + sinAsin²B – sinBsin²A = 0
⇒ (sinA – sinB) (sinB – sinC) (sinC – sinA) = 0
⇒ (sinA – sinB) = 0 or (sinB – sinC) = 0 or (sinC – sinA) = 0
⇒ A = B or B = C or C = A
a = 1 + 2 + 4 + ......... upto n terms
⇒ a = 2ⁿ – 1
b = 1 + 3 + 9 + ......... upto n terms
$\Rightarrow \mathrm{b}=\frac{3^{\mathrm{n}}-1}{2}$
c = 1 + 5 + 25 + ......... upto n terms
$\Rightarrow \mathrm{c}=\mathrm{b}=\frac{5^{\mathrm{n}}-1}{4}$
$\mathrm{\Delta }=\mid \begin{array}{ccc}\mathrm{a}& 2\mathrm{b}& 4\mathrm{c}\\ 2& 2& 2\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\mid$
$=\mid \begin{array}{ccc}{2}^{\mathrm{n}}-1& 3^{\mathrm{n}}-1& 5^{\mathrm{n}}-1\\ 2& 2& 2\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\mid$
$=\mid \begin{array}{ccc}{2}^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\\ 2& 2& 2\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\mid +\mid \begin{array}{ccc}-1& -1& -1\\ 2& 2& 2\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\mid$
$=0-2\mid \begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 2^{\mathrm{n}}& 3^{\mathrm{n}}& 5^{\mathrm{n}}\end{array}\mid =0$