1. The problem is to evaluate the integral $$\int_{}^{x} \frac{1}{\cos^2(t+\frac{\pi}{2})} \, dt$$. 2. Recall the trigonometric identity: $$\cos\left(\theta + \frac{\pi}{2}\right) = -\sin(\theta)$$. 3. Therefore, $$\cos^2\left(t + \frac{\pi}{2}\right) = \sin^2(t)$$. 4. Substitute into the integral: $$\int_{}^{x} \frac{1}{\cos^2(t+\frac{\pi}{2})} \, dt = \int_{}^{x} \frac{1}{\sin^2(t)} \, dt = \int_{}^{x} \csc^2(t) \, dt$$. 5. The integral of $$\csc^2(t)$$ is $$-\cot(t) + C$$. 6. Evaluating the definite integral from 0 to $$x$$: $$\int_0^{x} \csc^2(t) \, dt = [-\cot(t)]_0^{x} = -\cot(x) + \cot(0)$$. 7. Note that $$\cot(0)$$ is undefined, so the integral is improper at the lower limit 0. If the lower limit is unspecified, the indefinite integral is: $$\int \frac{1}{\cos^2(t+\frac{\pi}{2})} \, dt = -\cot(t) + C$$. Final answer: $$\boxed{-\cot\left(x + \frac{\pi}{2}\right) + C}$$ (rewriting in terms of the original variable shift).