Prove that - - 1 a a2- a a2 1- a2 1 a - - - 1 - a32-

[ #160; 1 amp; #160; a amp; #160; a^2; #160; a amp; #160; a^2 amp; #160; 1; #160; a^2 amp; #160; 1 amp; #160; a ] ...

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Properties of Determinants

Prove that ...

Question

Prove that $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} a & a^{2} & 1 a^{2} & 1 & a \end{matrix} ∣ ∣ ∣ ∣ ∣ = - {(1 - a^{3})}^{2} .$

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|\begin{matrix} 1 & b + c & b^{2} + c^{2} \\ 1 & c + a & c^{2} + a^{2} \\ 1 & a + b & a^{2} + b^{2} \end{matrix}| = (a - b) (b - c) (c - a)

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 b & b^{2} & 1 c & c^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ \neq 0

a b c = - 1

a_{0}, a_{1}, a_{2}, a_{3} . . . .

(1 + x + x^{2})^{n}

a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . + (- 1)^{n - 1} a_{n - 1}^{2} = \frac{1}{2} a_{n} {1 - (- 1)^{n} a_{n}}

a_{0}, a_{1}, a_{2}

^{2})^{n}

a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . . + (- 1)^{n - 1} a_{n - 1}^{2}

= \frac{1}{2} a_{n} (1 - (- 1)^{n} a_{n})

prove that 1 + sec 20 = cot 40 cot 30

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