The set of all real values of $\lambda$ for which the function 

$\mathrm{f}\left(\mathrm{x}\right)=\left(1-{\cos }^2\mathrm{x}\right)·(\lambda +\mathrm{sinx}),\mathrm{x}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right),\text{ has exactly one maxima and exactly one }$minima, is:

$\begin{array}{l}f\left(x\right)=\left(1-{Cos}^{2}x\right)(\lambda +\sin x) ={Sin}^{2}x(\lambda +\sin x)\\ f^1\left(x\right)={\sin }^{2}xCosx+(\lambda +\sin x)2\sin x\cos x\\ =\sin x\cos x(2\lambda +3\sin x)\end{array}$

For maximum or minimum,

$\begin{array}{l}{f}^1\left(x\right)=0 \Rightarrow \sin x=0\text{ or }\sin x=\frac{-2\lambda }{3}\\ \text{ As there is exactly one max. and exactly one min, }{f}^{\text{'}}\left(x\right)=0\text{ has exactly }2\text{ roots }\\ -1<\frac{2\lambda }{3}<1\text{ and }\frac{-2\lambda }{3}\ne 0 \Rightarrow \lambda \in \left(\frac{-3}{2},\frac{3}{2}\right)-\left\{0\right\}\end{array}$