tg(x)= ???? cos(x)

tan(x) = cos(x)  

sin(x)/cos(x) - cos(x) = 0

[sin(x) - cos²(x)] / cos(x)  = 0

[sinx - (1 - sin²(x)) ] / cos(x) = 0

Since the left side is only 0 if the numerator = 0 , we have

sin²(x) + sin(x) - 1 = 0

Let a = sin(x)    .....so we have

a² + a - 1 = 0

a² + a + 1/4 = 1 + 1/4

(a + 1/2)² = 5/4     take the square root of both sides

a + 1/2 = ±√5/2

a = [±√5 - 1]/2

sin(x) = [±√5 - 1]/2

So

sin(x) = -Phi      or  sin(x) = phi

The first is impossible since -Phi < -1

The second yields

sin^-1(phi) = sin^-1([√5 - 1]/2) ≈ 38.17°

This could also be a 38.17° in the second quadrant ≈ 141.83°

To convert cos(x) to tan(x) u just need to do this : 

x=acos(cos(x)) and then just tan x and u have y`r answer :)

Or you can draw the right angled triangle and get the exact value on the 3rd side using pythagoras and 'read' the cos off the triangle.

tg(x)= ???? cos(x)

$$tg{(x)} = \sqrt { \frac{1-\cos{(2x)}}{1+\cos{(2x)}}}$$

Example:

$$tg{(45)} = \sqrt { \frac{1-\cos{(90)}}{1+\cos{(90)}}} = \sqrt{\frac{1}{1}} = 1 \quad ok!$$