Question

transforme em produto y= 1+cosx

Answer 1

Identidades utilizadas:

$\begin{array}{rclc} \cos \alpha + \cos \beta&=&2\cdot \cos \left(\frac{\alpha+\beta}{2} \right )\cdot \cos \left(\frac{\alpha-\beta}{2} \right)&\;\;\;\text{(i)}\\ \\ \cos \left(-\alpha \right )&=&\cos \alpha&\;\;\;\text{(ii)} \end{array}$


Então,

$y=1+\cos x\\ \\ y=\cos 0+\cos x\\ \\ y=2\cdot \cos \left(\frac{0+x}{2} \right )\cdot \cos \left(\frac{0-x}{2} \right )\\ \\ y=2\cdot \cos \left(\frac{x}{2} \right )\cdot \cos \left(-\frac{x}{2} \right )\\ \\ y=2\cdot \cos \left(\frac{x}{2} \right )\cdot \cos \left(\frac{x}{2} \right )\\ \\ \boxed{y=2\cdot \cos^{2} \left(\frac{x}{2} \right )}$