Let E the expression of the left side of the equation:
[tex]E=\sec^6(x) (\sec (x)\tan (x)) - \sec^4(x) (\sec (x)\tan (x))\\\\
E=\sec^7(x)\tan(x)-\sec^5(x)\tan(x)\\\\
E=\sec^5(x)\tan(x)(\sec^2(x)-1)[/tex]
We must know that [tex]\tan^2(x)+1=\sec^2(x)\iff \sec^2(x)-1=\tan^2(x)[/tex]. So, using in the equation above:
[tex]E=\sec^5(x)\tan(x)(\underbrace{\sec^2(x)-1}_{\tan^2(x)})\\\\
E=\sec^5(x)\tan(x)(\tan^2(x))\\\\
\boxed{E=\sec^5(x)\tan^3(x)}[/tex]
Then, the equation is true.
