#### Answer

The primary step to solve the equation $4{{\cos }^{2}}x+4\sin x-5=0$ is to replace the trigonometric function of ${{\cos }^{2}}x$ with the $1-{{\sin }^{2}}x$.

#### Work Step by Step

So, in order to simplify the trigonometric functions and equations, we change cos into sin, so that all the terms become sin and it is easier to solve.
$\begin{align}
& 4{{\cos }^{2}}x+4\sin x-5=0 \\
& 4\left( 1-{{\sin }^{2}}x \right)+4\sin x-5=0
\end{align}$
Thus, the primary step to solve the equation $4{{\cos }^{2}}x+4\sin x-5=0$ is to replace the
trigonometric function of ${{\cos }^{2}}x$ with $1-{{\sin }^{2}}x$.