Answer:
To solve the equation sinθ(1+tanθ) = tanθ(sinθ+cosθ), let's simplify both sides step-by-step.
1. Expand the left side of the equation:
sinθ + sinθtanθ = tanθ(sinθ + cosθ)
2. Distribute tanθ on the right side:
sinθ + sinθtanθ = tanθsinθ + tanθcosθ
3. Move all terms to one side of the equation:
sinθ + sinθtanθ - tanθsinθ - tanθcosθ = 0
4. Factor out sinθ from the first two terms and factor out tanθ from the last two terms:
sinθ(1 + tanθ) - tanθ(sinθ + cosθ) = 0
5. Now, we can see that both sides of the equation are equal to zero:
sinθ(1 + tanθ) - tanθ(sinθ + cosθ) = 0
This equation is already simplified and cannot be further simplified.