Question

prove that 1-cos^2theta = sin^2theta​

Answer 1

Answer:

1 - cos²θ = sin²θ

Yes that's true

Proof :

Taking ∆ABC

AB is a side

BC is another side

AC is hypotenuse

With lengths

AB = x cm

BC = y cm

AC = ?

Using Pythagoras theorem

(Hypotenuse)² = (side)² + (side)²

AC² = AB² + BC²

AC² = x² + y²

AC = √x² + y²

We have:

1 - cos²θ = sin²θ

Considering angle C as 'θ'

We know that

cosθ = adjacent/hypotenuse

sinθ = opposite/hypotenuse

Here,

Opposite side of 'θ' = AB = x cm

Adjacent side of 'θ' = BC = y cm

Hypotenuse of triangle = AC = √x² + y²

Substituting we get

1 - (y/√x² + y²)² = (x/√x² + y²)²

1 - y²/x² + y² = x²/x² + y²

x² + y² - y²/x² + y² = x²/x² + y²

x²/x² + y² = x²/x² + y²

LHS = RHS

Hence proved