How can we derive and prove the double angle formulas for sine, cosine, and tangent starting from the definitions of these functions?

To derive the double angle formulas for sine, cosine, and tangent, we use the angle addition formulas. For sine, sin(2θ) = 2sin(θ)cos(θ). For cosine, cos(2θ) = cos²(θ) – sin²(θ), which can also be written as 2cos²(θ) – 1 or 1 – 2sin²(θ). For tangent, tan(2θ) = 2tan(θ) / (1 – tan²(θ)). These derivations rely on the fundamental trigonometric identities and properties.

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