The number of ordered triplets (A, B, C) satisfying the equation  $(2{\sin }^{2}A+2^{-{\cos }^{2}B})e^C=e^{2C}+e^C+1$  where $A,B\in [0,2\pi ]\text{ and }C\in R,$ is

$$\begin{gathered} 2{\sin }^{2}A+2^{-{\cos }^{2}B}\le 3 \\ {e}^c+\frac{1}{e^c}+1\ge 3 \end{gathered}$$

The equation will satisfy when  ${\sin }^{2}A=1,{\cos }^{2}B=0\text{ and }{e}^c=1$
 $\alpha =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\frac{3\pi }{2},B=\frac{\pi }{2},\frac{3\pi }{2}\text{ and }c=0$
Number of ordered triplets is 4