can't figure out the step using trigonometric property

Question

I have problem that I cannot figure out with differential equation and using trigonometric property. How did they go from a red circle to an orange circle that I drew ???

Answer 1

By the Pythagorean identity, $\cos^2(2x) + \sin^2(2x) = 1$. Dividing through by $\cos^2(2x)$ yields $1+\tan^2(2x) = \sec^2(2x)$.

Answer 2

$$\sin^2{2t} + \cos^2{2t} = 1 $$ Multiplying both sides by $\sec^2{2t}$ gives $$ \frac{\sin^2{2t}}{\cos^2{2t}} + 1 = \sec^2{2t} = 1 + \tan^2{2t}$$