This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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To prove the identity ( 3x - 3x)/(1 - 2 2x) = x + x, we will start with the left-hand side (LHS) and transform it into the right-hand side (RHS) using trigonometric identities. Step 1: Expand 3x and 3x using the triple angle formulas: 3x = 4^3 x - 3 x 3x = 3 x - 4^3 x Substitute these into the numerator of the LHS: Numerator = (4^3 x - 3 x) - (3 x - 4^3 x) Numerator = 4^3 x - 3 x - 3 x + 4^3 x Numerator = 4(^3 x + ^3 x) - 3( x + x) Step 2: Use the sum of cubes identity a^3 + b^3 = (a+b)(a^2 - ab + b^2) for ^3 x + ^3 x. ^3 x + ^3 x = ( x + x)(^2 x - x x + ^2 x) Since ^2 x + ^2 x = 1, this simplifies to: ^3 x + ^3 x = ( x + x)(1 - x x) Step 3: Substitute this back into the numerator expression from Step 1. Numerator = 4( x + x)(1 - x x) - 3( x + x) Factor out ( x + x): Numerator = ( x + x) [4(1 - x x) - 3] Numerator = ( x + x) [4 - 4 x x - 3] Numerator = ( x + x) [1 - 4 x x] Step 4: Simplify the denominator of the LHS using the double angle formula 2x = 2 x x. Denominator = 1 - 2 2x Denominator = 1 - 2(2 x x) Denominator = 1 - 4 x x Step 5: Combine the simplified numerator and denominator. ( 3x - 3x)/(1 - 2 2x) = (( x + x) [1 - 4 x x])/(1 - 4 x x) Assuming 1 - 4 x x ≠ 0, we can cancel the common term in the numerator and denominator. ( 3x - 3x)/(1 - 2 2x) = x + x This matches the RHS. Thus, the identity is proven. ( 3x - 3x)/(1 - 2 2x) = x + x That's 2 down. 3 left today — send the next one.