Question

I NEED HELP PLEASEEEEE

Answer 1

Answer:

Option (3)

Step-by-step explanation:

$\text{tan}(\frac{\theta}{2})=\text{sin}\theta$

$\text{tan}(\frac{\theta}{2})=2\text{sin}{\frac{\theta}{2}} \text{cos}(\frac{\theta}{2})$

$\frac{\text{sin}\frac{\theta}{2}}{\text{cos}\frac{\theta}{2}} =2\text{sin}{\frac{\theta}{2}}\text{cos}(\frac{\theta}{2})$

$\text{sin\frac{\theta}{2}}=2\text{sin}{\frac{\theta}{2}}\text{cos}^2(\frac{\theta}{2})$$\text{sin\frac{\theta}{2}}=2\text{sin}{\frac{\theta}{2}}\text{cos}^2(\frac{\theta}{2})$$\text{sin}\frac{\theta}{2}=2\text{sin}{(\frac{\theta}{2})}\text{cos}^2(\frac{\theta}{2})$

$\text{sin}\frac{\theta}{2}-2\text{sin}{(\frac{\theta}{2})}\text{cos}^2(\frac{\theta}{2})=0$

$\text{sin}\frac{\theta}{2}[1-2\text{cos}^2(\frac{\theta}{2})]=0$

$\text{sin}\frac{\theta}{2}=0$ ⇒ $\theta=0$

$1-2\text{cos}^2(\frac{\theta}{2})=0$

$\text{cos}(\frac{\theta}{2})=\frac{1}{\sqrt{2}}$

$\frac{\theta}{2}=\frac{\pi}{4}$

$\theta=\frac{\pi}{2}$

Therefore, θ = 0 and $\frac{\pi}{2}$ are the solutions.

Option (3) will be the answer.