If |sin−1xsin−1(2x)sin−1(3x)sin−1(3x)sin−1xsin−1(2x)sin−1(2x)sin−1(3x)sin−1x|=0 then number of values of satisfying the equation is equal to

|abcbcacab|=3abc(−a3−b3−c3)⇒|abccabbca|=a3+b3+c3−3abc a3+b3+c3−3abc=0⇒(a+b+c)(a2+b2+c2−ab−bc−ca)=0 ⇒ a+b+c=0 or a=b=c ⇒ sin−1x+sin−12x+sin−13x=0 or sin−1x=sin−12x=sin−13x α= sin−1x,β=sin−12x,γ=sin−13x⇒x=0 ⇒ α+β=−γ⇒cosαcosβ−sinαsinβ=cosγ ⇒ 1−x21−4x2−2x2=1−9x2⇒ only solution

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If $| \begin{matrix} \sin^{- 1} x & \sin^{- 1} (2 x) & \sin^{- 1} (3 x) \\ \sin^{- 1} (3 x) & \sin^{- 1} x & \sin^{- 1} (2 x) \\ \sin^{- 1} (2 x) & \sin^{- 1} (3 x) & \sin^{- 1} x \end{matrix} | = 0$ then number of values of satisfying the equation is equal to

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Detailed Solution

$| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} | = 3 a b c (- a^{3} - b^{3} - c^{3}) \Rightarrow | \begin{matrix} a & b & c \\ c & a & b \\ b & c & a \end{matrix} | = a^{3} + b^{3} + c^{3} - 3 a b c a^{3} + b^{3} + c^{3} - 3 a b c = 0 \Rightarrow (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = 0 \Rightarrow a + b + c = 0 o r a = b = c \Rightarrow \sin^{- 1} x + \sin^{- 1} 2 x + \sin^{- 1} 3 x = 0 o r \sin^{- 1} x = \sin^{- 1} 2 x = \sin^{- 1} 3 x α = \sin^{- 1} x, β = \sin^{- 1} 2 x, γ = \sin^{- 1} 3 x \Rightarrow x = 0 \Rightarrow α + β = - γ \Rightarrow \cos α \cos β - \sin α \sin β = \cos γ \Rightarrow \sqrt{1 - x^{2}} \sqrt{1 - 4 x^{2}} - 2 x^{2} = \sqrt{1 - 9 x^{2}} \Rightarrow only solution$