Arc-ed up

$\begin{aligned} \arcsin 1 &= \frac{\pi}{2}. \\ \arccos 1 &= 0. \\ \arctan 1 &= \frac{\pi}{4}. \\ \text{arccot} \, 1 &= \frac{\pi}{4}. \\ \text{arcsec} \, 1 &= 0. \\ \text{arccsc} \, 1 &= \frac{\pi}{2}. \end{aligned}$

We can confirm these by taking the trigonometric functions of these values to see if they get $1.$ Thus,

$\begin{aligned} \arcsin 1 + \arccos 1 + \arctan 1 + \text{arccot} \, 1 + \text{arcsec} \, 1 + \text{arccsc} \, 1 &= \frac{\pi}{2} + 0 + \frac{\pi}{4} + \frac{\pi}{4} + 0 + \frac{\pi}{2} \\ &= \boxed{\frac{3\pi}{2}}. \end{aligned}$

$\text{arcsin}1=\frac{\pi}{2}$

$\text{arccos}1=0$

$\text{arctan}1=\frac{\pi}{4}$

$\text{arccot}1=\frac{\pi}{4}$

$\text{arcsec}1=0$

$\text{arccsc}1=\frac{\pi}{2}$

$\arcsin 1 + \arccos 1 + \arctan 1 + \text{arccot} \, 1 + \text{arcsec} \, 1 + \text{arccsc} \,1 =\boxed{\frac{3\pi}{2}}$