TRIGONOMETRY: How to prove trigonometry indentity by Pythagoras Theorem?

We use different identity in trigonometry to solve the problems. We may prove this identity with the help of Pythagoras Theorem.


Pythagoras Theorem

Pythagoras theorem says-

In a right angle triangle square of the larger side(hypotenuse) is equal to the sum of the squares of the other two sides(Perpendicular & Base).

H² = P² + B²
Where, H = Hypotenuse, P = Perpendicular (opposite to angle θ) & B = Base.

1.) How to prove sin²θ+cos²θ = 1 with the help of Pythagoras Theorem?

Solve-

We know that H² = P² + B²
And according to the above picture we have-

AC² = BC² + AB²

To prove the identity we will divide the above expression by AC² (Because we want to make 1 to AC in the above expression) 

AC²/AC² = BC²/AC² + AB²/AC²

So as per figure & trigonometry, we will get-

1 = Sin²θ + Cos²θ {BC/AC = Sinθ, AB/AC = Cosθ}

Hence Proved

2.) How to prove 1 + tan²θ = sec²θ with the help of Pythagoras Theorem?

Solve-

We know that H² = P² + B²
And according to the above picture we have-

AC² = BC² + AB²

To prove the identity we will divide the above expression by AB² (Because we want to make 1 to AB in the above expression)

AC²/AB² = BC²/AB² + AB²/AB²

So as per figure & trigonometry, we will get-

sec²θ = tan²θ + 1 {AC/AB = secθ, BC/AB = tanθ}
Or
tan²θ + 1 = sec²θ

Hence Proved

3.) How to prove cot²θ + 1 = cosec²θ with the help of Pythagoras Theorem?

Solve-

We know that H² = P² + B²
And according to the above picture we have-

AC² = BC² + AB²

To prove the identity we will divide the above expression by BC² (Because we want to make 1 to BC in the above expression)

AC²/BC² = BC²/BC² + AB²/BC²

So as per figure & trigonometry, we will get-

cosec²θ =1 + cot²θ  {AC/BC = cosecθ, AB/BC = cotθ}
Or
1 + cot²θ = cosec²θ

Hence Proved