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?
We know that H² = P² + B²
And according to the above picture we have-
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
Or
tan²θ + 1 = sec²θ
Hence Proved
3.) How to prove cot²θ + 1 = cosec²θ with the help of Pythagoras Theorem?
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