Podemos reescrever a função $f(x)$ como $$f(x) = \left(\cos x + \dfrac{\sin x \sin \frac{x}{2}}{\cos \frac{x}{2}}\right) = \left(\cos x + \dfrac{2 \sin \frac{x}{2} \cos\frac{x}{2} \sin \frac{x}{2}}{\cos \frac{x}{2}}\right) = \\ = \left(\cos x + 2 \sin^2 \frac{x}{2}\right) = \left(1 - 2 \sin^2 \frac{x}{2} + 2 \sin^2 \frac{x}{2}\right) = 1 \ \text{(constante)}.$$