Animated Solution for Mathematics - Trigonometry: If $5({\tan }^{2}x-{\cos }^{2}x)=2\cos 2x+9$, then the value of $\cos 4x$ is:

The given trigonometric equation is:

Our objective is to determine the value of $\cos 4x$. To simplify the expression, we will unify the terms by expressing everything in terms of ${\cos }^{2}x$.

First, we dismantle the ${\tan }^{2}x$ term using the identity ${\tan }^{2}x={\sec }^{2}x-1$. Since $\sec x=\frac{1}{\cos x}$, we rewrite this as:

Next, we apply the double angle identity for $\cos 2x$:

Substituting these into the original equation, we obtain:

To simplify the algebra, let $t={\cos }^{2}x$. The equation transforms into:

Simplifying the right-hand side gives $4t+7$. Multiplying the entire equation by $t$ to eliminate the fraction, we get:

Expanding and rearranging all terms to one side yields the quadratic equation:

We factor the quadratic $9t^2+12t-5=0$ by splitting the middle term:

This provides two potential values: $t=-\frac{5}{3}$ or $t=\frac{1}{3}$. However, since $t={\cos }^{2}x$, it must satisfy the constraint $t\ge 0$.

We reject $t=-\frac{5}{3}$. Thus, the only valid solution is:

First, we calculate $\cos 2x$ using the identity $\cos 2x=2{\cos }^{2}x-1$:

Finally, we apply the double angle identity for $\cos 4x$:

Substituting $\cos 2x=-\frac{1}{3}$ into the equation:

The final result is: