Now we will study the next part of trigonometry.

###
**1) Fundamental Identities:**

**a)**sin

^{2}Θ + cos

^{2}Θ = 1

**b)**sec

^{2}Θ - tan

^{2}Θ = 1

^{}

**c)**csc

^{2}Θ - cot

^{2}Θ = 1

^{}

### Proof:

###
** a) **sin^{2} Θ + cos^{2} Θ = 1

Here,

1) x

^{2}+ y^{2}= r^{2}-----------------(1)
Dividing by equation 1 by r

^{2},we get,
2) (x

^{2 }/ r^{2}+ y^{2 }/ r^{2}= r^{2 }/ r^{2}
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 Θ)

^{2}+ (sin Θ)^{2}= 1 ---(3)
6) (sin Θ)

^{2}+ (cos Θ)^{2}= 1###
** b) **sec^{2} Θ - tan^{2} Θ = 1

Here,

1) x

^{2}+ y^{2}= r^{2}-------------------------(1)
Dividing by equation 1 by x

^{2},we get,
2) (x

^{2 }/ x^{2}+ y^{2 }/ x^{2}= r^{2 }/ x^{2}
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) (1)

^{2}+ (tan Θ)^{2}= (sec Θ)^{2}---(3)
6) (sec Θ)

^{2}- (tan Θ)^{2}= 1###
**c) **csc^{2} Θ - cot^{2} Θ = 1

Here,

1) x

^{2}+ y^{2}= r^{2}-------------------------(1)
Dividing by equation 1 by y

^{2},we get,
2) (x

^{2 }/ y^{2}+ y^{2 }/ y^{2}= r^{2 }/ y^{2}
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 Θ)

^{2}+ (1)^{2}= (csc Θ)^{2}---(3)
6) (csc Θ)

^{2}- (cot Θ)^{2}= 1