how to prove Pythagorean Identities

Question :  how to prove Pythagorean Identities?

 Solution :

There are three Pythagorean Identities

• sin²𝜃 + cos²𝜃 = 1
•  1+ tan²𝜃 = sec²𝜃
• 1+ cot²𝜃 = csc² 𝜃

Proof :

Let a right triangle on the XY plane is constructed as follows

• To prove sin²𝜃 + cos²𝜃 = 1

Coordinates of A are (x,0) and B are (0,y)

So the distance AB is  r=√(x^2+y^2)

Therefore r^2 = x^2+y^2

In the triangle AOB

sin 𝜃 = y/r , cos 𝜃 = x/r

Squaring and adding these sin²𝜃 + cos²𝜃 =(x^2+y^2)/r^2  =r^2/r^2 =1

Therefore sin²𝜃 + cos²𝜃 = 1

Tp prove 1+ tan²𝜃 = sec²𝜃

Divide  sin²𝜃 + cos²𝜃 = 1 by cos²𝜃

  tan²𝜃 +1= 1/cos²𝜃 = sec²𝜃

hence  1+ tan²𝜃 = sec²𝜃

• To prove 1+ cot²𝜃 = csc² 𝜃

Divide  sin²𝜃 + cos²𝜃 = 1 by sin²𝜃

  1+cot²𝜃 = 1/sin²𝜃 = cosec²𝜃

hence  1+ cot²𝜃 = csc² 𝜃