There are eight fundamental relationships. These relationships, among the trigonometric functions, are extremely important in more advanced courses in mathematics. They can be divided into three categories: reciprocal, ratio, and pythagorean.
Reciprocal Relationships
sin(x) = 1/csc(x) [opp/hyp]
cos(x) = 1/sec(x) [adj/hyp]
cot(x) = 1/tan(x) [opp/adj]
Ratio Relationships
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)
Pythagorean Relationships
sin2(x) + cos2(x) = 1
tan2(x) + 1 = sec2(x)
cot2(x) + 1 = csc2(x)
Example 1: Simplify cos A * tan A, when cos A ≠ 0.
Essentially, we want everything in terms of sin A and cos A
cos A * tan A = cos A * (sin A/cos A)
The cos A can be cancelled out.
cos A * (sin A/cos A) = sin A
Example 2: Simplify (tan A)^2 * (cos A)^2 * (csc A)
Again, we want everything in terms of sin A and cos A
(tan A)^2 * (cos A)^2 * (csc A) = (sin A)^2/(cos A)^2 * (cos A)^2 * (1/Sin A)
Cancel out terms in the numerator and denominator.
