cos(asin(1/x))

draw a right angled triangle.

mark one of the acute angles $$\theta$$

$$\theta$$ = asin (x/1)

now

$$sin\theta = \frac{opp}{hyp}=\frac{1}{x}$$

 

put 1 on the opp side and x on the hypotenuse

find the adjacent side using the pythagorean theorem.

so you can read cos $$\theta$$ off the triangle.

BUT

You should consider the possible quadrants

if x is positive $$\theta$$ will be in the first quad so cos$$\theta$$ is also positive.

if x is negative $$\theta$$ will be in the 4ht quad so cos$$\theta$$ will be positive also.