If a2+b2+c2=1, then a2+b2+c2cos⁡θab(1−cos⁡θ)ba(1−cos⁡θ)b2+c2+a2cos⁡θca(1−cos⁡θ)cb(1−cos⁡θ) ac(1−cos⁡θ)bc(1−cos⁡θ)c2+a2+b2cos⁡θ equals

First multiply C1 by a,C2 by b,C3 by c, followed by multiplying R1 by 1/a,R2 by 1/b and R3 by 1/c we getΔ=a2+b2+c2cos⁡θb2(1−cos⁡θ)a2(1−cos⁡θ)b2+c2+a2cos⁡θa2(1−cos⁡θ)b2(1−cos⁡θ) c2(1−cos⁡θ)c2(1−cos⁡θ)c2+a2+b2cos⁡θ Using C1→C1+C2+C3 and a2+b2+c2=1 we getΔ=1b2(1−cos⁡θ)c2(1−cos⁡θ)1b2+1−b2cos⁡θc2(1−cos⁡θ)1b2(1−cos⁡θ)c2+1−c2cos⁡θUsing R2→R2−R1,R3→R3−R1, we getΔ=1b2(1−cos⁡θ)c2(1−cos⁡θ)0cos⁡θ000cos⁡θ=cos2⁡θ

If a2+b2+c2=1, then a2+b2+c2cos⁡θab(1−cos⁡θ)ba(1−cos⁡θ)b2+c2+a2cos⁡θca(1−cos⁡θ)cb(1−cos⁡θ) ac(1−cos⁡θ)bc(1−cos⁡θ)c2+a2+b2cos⁡θ equals

First multiply C1 by a,C2 by b,C3 by c, followed by multiplying R1 by 1/a,R2 by 1/b and R3 by 1/c we getΔ=a2+b2+c2cos⁡θb2(1−cos⁡θ)a2(1−cos⁡θ)b2+c2+a2cos⁡θa2(1−cos⁡θ)b2(1−cos⁡θ) c2(1−cos⁡θ)c2(1−cos⁡θ)c2+a2+b2cos⁡θ Using C1→C1+C2+C3 and a2+b2+c2=1 we getΔ=1b2(1−cos⁡θ)c2(1−cos⁡θ)1b2+1−b2cos⁡θc2(1−cos⁡θ)1b2(1−cos⁡θ)c2+1−c2cos⁡θUsing R2→R2−R1,R3→R3−R1, we getΔ=1b2(1−cos⁡θ)c2(1−cos⁡θ)0cos⁡θ000cos⁡θ=cos2⁡θ

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If $a^{2} + b^{2} + c^{2} = 1$ , then $|\begin{array}{cc} a^{2} + (b^{2} + c^{2}) \cos θ & a b (1 - \cos θ) \\ b a (1 - \cos θ) & b^{2} + (c^{2} + a^{2}) \cos θ \\ c a (1 - \cos θ) & c b (1 - \cos θ) \end{array} \begin{matrix} a c (1 - \cos θ) \\ b c (1 - \cos θ) \\ c^{2} + (a^{2} + b^{2}) \cos θ \end{matrix}|$ equals

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$\sin^{2} θ$

$\cos^{2} θ$

answer is A.

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Detailed Solution

First multiply $C_{1}$ by $a, C_{2}$ by $b, C_{3}$ by $c,$ followed by multiplying $R_{1}$ by $1 / a, R_{2}$ by $1 / b$ and $R_{3}$ by $1 / c$ we get

$Δ = |\begin{array}{cc} a^{2} + (b^{2} + c^{2}) \cos θ & b^{2} (1 - \cos θ) \\ a^{2} (1 - \cos θ) & b^{2} + (c^{2} + a^{2}) \cos θ \\ a^{2} (1 - \cos θ) & b^{2} (1 - \cos θ) \end{array} \begin{matrix} c^{2} (1 - \cos θ) \\ c^{2} (1 - \cos θ) \\ c^{2} + (a^{2} + b^{2}) \cos θ \end{matrix}|$

Using $C_{1} \to C_{1} + C_{2} + C_{3}$ and $a^{2} + b^{2} + c^{2} = 1$ we get

$Δ = |\begin{array}{ccc} 1 & b^{2} (1 - \cos θ) & c^{2} (1 - \cos θ) \\ 1 & b^{2} + (1 - b^{2}) \cos θ & c^{2} (1 - \cos θ) \\ 1 & b^{2} (1 - \cos θ) & c^{2} + (1 - c^{2}) \cos θ \end{array}|$