We know:

sin x = (1 - cos^ x)^1/2

Therefore when cos x = 0.25

sin x = (1 - 0.25^2)^1/2

= (1 - 0.0625)^1/2

= 0.9375^1/2

= 0.968246

tan x = (sin x)/(cos x)

Substituting above values of sin x and cos x we get:

tan x = (0.9375^1/2)/0.05 = 3.87298

To calculate the value of the function tan x, we'll use the formula:

(tan x)^2 + 1 = 1/(cos x)^2

(tan x)^2 = 1/(cos x)^2 - 1

tan x = +/-sqrt [1/(cos x)^2 - 1]

tan x = +/- sqrt (16-1)

tan x = +/- sqrt15

We'll calculate the value of sin x, using the formula:

tan x = sin x / cos x

sin x = tan x * cos x

sin x = (+/- sqrt15) * (1/4)

The values of tanx and sin x could be positive or negative, depending on the quadrant where the angle x is located.

Here we use the relations (sin x)^2+(cos x)^2=1 and tan x =sinx/cosx.

First let's find find sin x.

(sin x)^2=1-(cos x)^2 = 1-0.25^2 =0.9375

**sin x = sqrt(0.9375)= 0.9682**

Now, tan x =sinx/cosx

=> **tan x =0.9682/0.25= 3.8729**

(All values given above are for values of x in the first quadrant, in the fourth quadrant the magnitude remains the same but the sign of sign x and tan x are negative)

cosx = 0.25.

To find sinx and tanx.

We use the trigonometric identity, cos^2x+sin^2 = 1

So sinx = +sqrt(1-cos^2), or -sqrt(1-cos^2x)

Therefore sinx = sqrt[1-.(25)^2] = sqrt (1-.0625) = sqrt(0.9375) = 0.968245836 in first quadrant.

sinx = -0.968245836 in the 4th quadrant, where cosx = 0.25

So tanx = sinx/cosx = 0.968245836/0.25 = 3.872983345 in 1st quadrant, where cosx = 0.25. Or

tanx = -3.872983345 in the 4th quadrant where cosx = 0.25