For 0 is less than or equal to theta comma theta is less than 2 pi , how many solutions are there to tangent of open paren theta over 2 close paren is equal to sine of theta ? Note: Do not include values that are undefined for tangent or sine of theta

Let x = tan(θ/2), which is defined throughout the interval except at θ = π. Using sin θ = 2x/(1 + x²), the equation becomes x = 2x/(1 + x²), so x = 0 or x² = 1. These correspond to θ = 0, π/2, and 3π/2; all are in the stated interval and make the original functions defined. Therefore, there are 3 solutions, choice D.

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