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Properties of Determinants

If A, B and C...

Question

If A, B and C are the angles of a triangle, then
$∣ ∣ ∣ ∣ \begin{matrix} s i n 2 A & s i n C & s i n B s i n C & s i n 2 B & s i n A s i n B & s i n A & s i n 2 C \end{matrix} ∣ ∣ ∣ ∣ =$

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Solution

$A, B a n d c a r e t h e a n g l e s o f a t r i a n g l e$

$L . H . S = ∣ ∣ ∣ ∣ \begin{matrix} s i n 2 A & s i n C & s i n B s i n C & s i n 2 B & s i n A s i n B & s i n A & s i n 2 C \end{matrix} ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ \begin{matrix} 2 k a c o s A & k c & k b k c & 2 k b c o s B & k a k b & k a & 2 k c c o s C \end{matrix} ∣ ∣ ∣ ∣ (F r o m s i n e r u l e)$

$= k^{3} ∣ ∣ ∣ ∣ \begin{matrix} 2 a c o s A & c & b c & 2 b c o s B & a b & a & 2 c c o s C \end{matrix} ∣ ∣ ∣ ∣$

$= k^{3} ∣ ∣ ∣ ∣ \begin{matrix} a c o s A + a c o s A & a c o s B + b c o s A & a c o s C + c c o s A a c o s B + b c o s A & b c o s B + b c o s B & b c o s C + c c o s B a c o s C + c c o s A & b c o s C + c c o s B & c c o s C + c c o s C \end{matrix} ∣ ∣ ∣ ∣ (f r o m p r o j e c t i o n r u l e)$

$= k^{3} ∣ ∣ ∣ ∣ \begin{matrix} c o s A & a & 0 c o s B & b & 0 c o s C & c & 0 \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ \begin{matrix} a & c o s A & 0 b & c o s B & 0 c & c o s C & 0 \end{matrix} ∣ ∣ ∣ ∣$

$= 0$

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