Prove that - a -1 a -1 - b -1 - a -1 a -1 - b -1 - 2 b 2 b 2 - a 2-

a ^ 1 a ^ 1 + b ^ 1 + a ^ 1 a ^ 1 b ^ 1 = 2 b ^ 2 b ^ 2 a ^ 2 .Let us consider, #160; a=a^ 1a^ 1+b^ 1+a^ 1a^ 1 b^ 1 #160; #160; ...

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Properties of Iota

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Prove that : $\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{2 b^{2}}{b^{2} - a^{2}}$ .

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\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{2 b^{2}}{b^{2} - a^{2}}

x = 1 + a + a^{2} + . . . . .

y = 1 + b + b^{2} . . . .

| a | < 1

| b | < 1

1 + a b + a^{2} b^{2} + . . . . = \frac{x y}{x + y - 1}

a \int a f (x) d x = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 a \int 0 f (x) d x & if f (x) is an even function 0 & if f (x) is an odd function \end{matrix}

1 \int - 1 {sin}^{5} x {cos}^{4} x d x

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

a, b, c

a^{2} = b^{2} + c^{2} - 2 b c \sqrt{1 - a^{2}}, b^{2} = c^{2} + a^{2} - 2 c a \sqrt{1 - b^{2}}, c^{2} = a^{2} + b^{2} - 2 a b \sqrt{1 - c^{2}}

a = c \sqrt{1 - b^{2}} + b \sqrt{1 - c^{2}}

If $X = 1 + a + a^{2} + . . . . \infty$ , where |a| < 1 and $y = 1 + b + b^{2} + . . . . . \infty$ where |b| <1,

prove that $1 + a b + a^{2} b^{2} + . . . . . \infty \frac{x y}{x + y - 1}$

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