Unit circle, axis, and complementary (orthogonal) axis designated as coaxis. 
Secant line, a geometric object, cuts the circle at an angle of theta to the horizontal axis. The secant line intersects the circle at point P. 
The same secant line also determines the complementary angle to theta (another geometric object), denoted cotheta. 
The length of the green line, dropping from P to the axis, measures the sine of theta: opposite side of a right triangle over a hypotenuse of the unit circle. 
The pink line is a geometric line tangent to the unit circle at (1,0) on the horizontal axis. 
The length of the red line, intercepted by the secant line along the tangent line, measures the tanget of theta. 
The length of the blue line, intercepted by the tangent line along the secant line, measures the secant of theta. 
The length of the green line, dropping from P to the coaxis, measures the sine of cotheta: opposite side of a right triangle over a hypotenuse of the unit circle. Hence, cosine of theta. 
The pink line is a geometric line tangent to the unit circle at (1,0) on the coaxis. 
The length of the red line, intercepted by the secant line along the tangent line, measures the tanget of cotheta. Hence, cotanget of theta 
The length of the blue line, intercepted by the tangent line along the secant line, measures the secant of cotheta. Hence, cosecant of theta. 
The three functions of theta measuring sine, tangent, and secant shown together. 
The three functions of cotheta measuring sine of cotheta (cosine of theta), tangent of cotheta (cotangent of theta), and secant of cotheta (cosecant of theta). 
All six functions of theta are shown in this image. The representations for secant lies on top of that for cosecant of theta. The latter is thus shaded a lighter shade. 

A number of trigonmetric identities are evident from this visual approach.
From the Pythagorean Theorem, it follows that:
sin^{2} theta + cos^{2}
theta = 1, the radius of the unit circle measured along the secant line;
sec^{2} theta = tan^{2}
theta + 1, the radius of the unit circle measured along the horizontal
axis;
csc^{2} theta = cot^{2}
theta + 1, the radius of the unit circle measured along the coaxis.
What others do you note?