Pythagorean trigonometric identity, list with Proof

The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In the shown Right Angled Triangle, we can write,

By Pythagoras theorem:

AB² + BC² =AC²                  Eq (1)   (Where AB is Base, BC is Perpendicular & AC is the Hypotenuse  of the Right Angled Triangle shown)

Let us divide the above equation Eq (1) by AC² on both sides, We get:

AB2	+	BC2	=	AC2
AC2		AC2		AC2

      Eq (2)    
As per the Right Angled Triangle, we know that

1	=	SinӨ	=	Length of Opposite side	=	BC
CosecӨ				Length of Hypotenuse		AC
1	=	CosӨ	=	Length of Adjacent side	=	AB
SecӨ				Length of Hypotenuse		AC
1	=	TanӨ	=	Length of Opposite side	=	BC
CotӨ				Length of Adjacent side		AB

So we get from the above equation Eq (2),

Cos²Ө +Sin²Ө = 1         Eq (2)

Now Let us divide the above equation Eq (1) by AB² on both sides, We get:

AB2	+	BC2	=	AC2
AB2		AB2		AB2

We get:

1 + Tan²Ө = Sec² Ө       Eq (3)

Similarly, on divide the above equation Eq (1) by BC² on both sides, We get:

AB2	+	BC2	=	AC2
BC2		BC2		BC2

 We get:

cot²Ө + 1 = Cosec² Ө      Eq (4)

Now Writing All Pythagorean identities (Eq (2), Eq (3)& Eq (4))”

Cos²Ө + Sin²Ө = 1

1 + Tan²Ө = Sec² Ө

cot²Ө + 1 = Cosec² Ө

These three identities  sometimes called the fundamental Pythagorean trigonometric identity.