If 1+sin2 θ = 3sin θ cos θ, then prove that tan θ = 1 or tan θ = 1/2

Sure, let's solve this step by step:

Step 1: Rewrite the given equation The given equation is 1 + sin2 θ = 3sin θ cos θ. We can rewrite this as sin2 θ - 3sin θ cos θ + 1 = 0.

Step 2: Use the identity sin2 θ = 1 - cos2 θ Substitute sin2 θ with 1 - cos2 θ in the equation. This gives us 1 - cos2 θ - 3sin θ cos θ + 1 = 0.

Step 3: Simplify the equation Simplify the equation to get 2 - cos2 θ - 3sin θ cos θ = 0.

Step 4: Use the identity sin θ = cos(π/2 - θ) Substitute sin θ with cos(π/2 - θ) in the equation. This gives us 2 - cos2 θ - 3cos(π/2 - θ) cos θ = 0.

Step 5: Use the identity cos(π/2 - θ) = sin θ Substitute cos(π/2 - θ) with sin θ in the equation. This gives us 2 - cos2 θ - 3sin θ cos θ = 0.

Step 6: Rewrite the equation in terms of tan θ We can rewrite the equation in terms of tan θ using the identity tan θ = sin θ / cos θ. This gives us 2 - (1 + tan2 θ) - 3tan θ = 0.

Step 7: Simplify the equation Simplify the equation to get tan2 θ + 3tan θ - 2 = 0.

Step 8: Solve the quadratic equation Solve the quadratic equation to get tan θ = 1 or tan θ = -2. However, since tan θ is positive in the given context, we discard the solution tan θ = -2.

Therefore, the solution to the given equation is tan θ = 1 or tan θ = 1/2.