1. ## inverse

For -infinity < t < +infinity

cos(arctan(t))

need some direction for this bad johnny...thanks

2. Try to express $\cos t$ in terms of $\tan t,$ can you do that?

After that evaluate $\cos(\arctan t).$

3. Are you trying to find the inverse of: $y=cos(arctan(t))$? If so, simply switch your y's and t's: $t=cos(arctan(y))$, and then solve for y.