Final answer:
The equation tanA(1+cos2A)=sin2A is proved by expanding the left-hand side using the double angle identity for cosine, simplifying, and recognizing that the final expression matches the double angle formula for sine, which is the right-hand side of the equation.
Step-by-step explanation:
To prove that tanA(1+cos2A)=sin2A, we can start by expanding and simplifying the left-hand side of the equation using trigonometric identities.
Using the double angle identity, we know that cos2A = cos²A - sin²A which can also be written as 1 - 2sin²A (since cos²A + sin²A = 1).
The equation becomes tanA(1 + 1 - 2sin²A), which simplifies to tanA(2 - 2sin²A). This simplifies further to 2tanA(1 - sin²A).
Because tanA = sinA/cosA, we can substitute to get 2(sinA/cosA)(cos²A). Simplifying this expression gives us 2sinAcosA, which is the right-hand side of the equation and is known to be the double angle formula for sine, sin2A.
Therefore, we have proved that tanA(1+cos2A)=sin2A.