To simplify the equation 2tanα/(1-tan2α) = 1 - 2√2, we can start by substituting the given value for tanα: 2√2tanα = 1 - 2√2(1 - tan2α) Next, we can expand the right side of the equation: 2√2tanα = 1 - 2√2 + 2√2tan2α Now, let's move all the terms to one side of the equation: 2√2tanα - 2√2tan2α = 1 - 2√2 Factor out common terms: 2√2(tanα - tan2α) = 1 - 2√2 Divide both sides by 2√2: tanα - tan2α = (1 - 2√2)/(2√2) Simplify the right side: tanα - tan2α = (1/√2 - 1) Since tanα = sinα/cosα and tan2α = sin2α/cos2α, we can rewrite the equation as: sinα/cosα - sin2α/cos2α = (1/√2 - 1) To simplify further, let's find a common denominator for the terms: (cos2α*sinα - sin2α) / cos2α = (1/√2 - 1) Using the identity cos2α + sin2α = 1, we can substitute cos2α with 1 - sin2α: (sinα*(1-sin2α) - sin2α) / (1 - sin2α) = (1/√2 - 1) Distribute the sinα: sinα - sin3α - sin2α = (1/√2 - 1)(1 - sin2α) Combine like terms: sinα(1 - sin2α) - sin2α = (1/√2 - 1)(1 - sin2α) Simplify the left side: sinα - sin3α - sin2α = (1/√2 - 1)(1 - sin2α) Now, let's distribute the (1/√2 - 1): sinα - sin3α - sin2α = 1/√2 - 1 - (1/√2 - 1)sin2α Simplify further: sinα - sin3α - sin2α = 1/√2 - 1 - 1/√2 + 1)(1 - sin2α) Combine like terms: sinα - sin3α - sin2α = 1/√2 - 1/√2 + 1 - sin2α Simplify the right side: sinα - sin3α - sin2α = 1 - sin2α Now, let's cancel out the sin2α terms from both sides: sinα - sin3α = 1 This is the simplified form of the equation.

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