Calculate the value of the sum sin(arcsin(1/4))+cos(2arccos(1/4)).

We'll start from the fact that sin(arcsin x) =
x.

Comparing, we'll get:

sin
(arcsin(1/4)) = 1/4

We'll note arccos(1/4) =
a

cos (2arccos(1/4)) = cos
2a

We'll apply the double angle
identity:

cos 2a = 2(cos a)^2 -
1

If a = arccos(1/4) => (cos a)^2 = (cos
arccos(1/4))^2 = 1/4^2

The sum will
become:

S = 1/4 + 2/4^2 - 1

S
= 1/4 + 2/16 - 1

S =
(4+2-16)/16

S = -10/16

S =
-5/8

The value of the trigonometric sum
sin(arcsin(1/4))+cos(2arccos(1/4)) = -5/8.