Besides sin, cos and tan, we also define 3 other trigonometric functions, namely, cot, sec and csc. We explore their properties here:
cotx*tanx=1 for all x where the product is defined
secx*cosx=1
cscx*sinx=1
Now, we will go through another fundamental trigonometric property of these 6 functions, namely the Pythagorean Identities.
1) (sinA)^2+(cosA)^2=1 for all real angles A.
In the right-angled triangle ABC, let the lengths of AB, BC and CA be c, a and b respectively.
By right-angled trigonometry in triangle ABC,
a=c*sinA, b=c*cosA
By the Pythagoras Theorem, we know that a^2+b^2=c^2
By the substitution above, we have (c*sinA)^2+(ccosA)^2=c^2. Dividing throughout by c^2 gives (sinA)^2+(cosA)^2=1.
Of course, this only works for A between 0 to 180 degrees. To extend the definition, refer to the Addition and Subtraction Formulas of sine and cosine. Also, note that sin0=0, sin90=1 and sin180=0. Similarly, cos0=1, cos90=0 and cos180=1. For more details on how to find cos0, sin0, cos180 and sin180, refer to the page on limits and continuity.
To obtain the other 2 formulas, we divide the equation by (sinA)^2 and (cosA)^2 respectively. We then obtain 1+(cotA)^2=(cscA)^2 and (tanA)^2+1=(secA)^2.
