To find the value of cos 0+tan x, we can use the Pythagorean theorem to find the length of the sides of triangle BDE and triangle CDE.

In triangle BDE, we have:

DE = 8 cm (given)

BE = 6 cm (given)

BD = ?

We can use the Pythagorean theorem to find the length of BD:

BD = √(DE^2 - BE^2) = √(8^2 - 6^2) = √(64 - 36) = √(28) = 2√(7) cm

In triangle CDE, we have:

DE = 8 cm (given)

CE = 4 cm (half of CD)

CD = ?

We can use the Pythagorean theorem to find the length of CD:

CD = √(DE^2 - CE^2) = √(8^2 - 4^2) = √(64 - 16) = √(48) = 4√(3) cm

Now that we have the lengths of the sides of the triangles, we can find the value of cos 0+tan x.

First, we can find the value of cos 0:

cos 0 = BD/DE = (2√(7))/8 = √(7)/4

Then, we can find the value of tan x:

tan x = BD/CE = (2√(7))/(4) = √(7)/2

Finally, we can add the value of cos 0 and tan x to find the value of cos 0+tan x:

cos 0+tan x = (√(7)/4) + (√(7)/2) = (√(7)/4) + (2√(7)/4) = 3√(7)/4

Therefore, the value of cos 0+tan x is 3√(7)/4.