## Thursday, August 28, 2014

### 81-Basics of Trigonometry - 09 Important key points

Now we will study the next part of trigonometry.

### 1) Fundamental Identities:

a) sin2 Θ + cos2 Θ = 1
b) sec2 Θ - tan2 Θ = 1
c) csc2 Θ - cot2 Θ = 1

### a) sin2Θ + cos2Θ = 1

Here,
1)  x2 + y2  = r2  -----------------(1)
Dividing by equation 1 by r2 ,we get,
2)  (x/ r2 + y/ r2 = r/ r2
3)  (x / r) 2 + (y / r) 2 = 1 -------(2)
4) We know that,
y/r = sin Θ
x/r  =  cos Θ, ----- from equation 2, we have,
5)  (cos Θ2 + (sin Θ2 = 1 ---(3)
6)  (sin Θ2 + (cos Θ2 = 1

### b) sec2Θ - tan2Θ = 1

Here,
1)  x2 + y2  = r2  -------------------------(1)
Dividing by equation 1 by x2 ,we get,
2)  (x/ x2 + y/ x2 = r/ x2
3)  (x / x) 2 + (y / x) 2 = (r / x) 2 -------(2)
4) We know that,
y/x =  tan Θ
r/x =  sec Θ , ----- from equation 2, we have,
5)  (12 + (tan Θ2 =  (sec Θ2 ---(3)
6)  (sec Θ2 - (tan Θ2 = 1

### c) csc2Θ - cot2Θ = 1

Here,
1)  x2 + y2  = r2  -------------------------(1)
Dividing by equation 1 by y2 ,we get,
2)  (x/ y2 + y/ y2 = r/ y2
3)  (x / y) 2 + (y / y) 2 = (r / y) 2 -------(2)
4) We know that,
x/y =  cot Θ
r/y =  csc Θ , ----- from equation 2, we have,
5)  (cot Θ2 + (12 =  (csc Θ2 ---(3)
6)  (csc Θ2 - (cot Θ2 = 1