Question

show that Cos 2 θ = 2 cos^2 θ - 1​

Answer 1

☆Answer☆

To Prove:-

Cos(2θ) = 2Cos²θ – 1

Solution:-

Cos(2θ) = Cos(θ+θ)

We have a identity,

Cos(A+B) = CosA.CosB – SinA.SinB

Now,

Cos(θ+θ) = CosθCosθ – SinθSinθ

Cos2θ = Cos²θ - Sin²θ

Using an identity, we have

Sin²θ+Cos²θ = 1

Sin²θ = 1-Cos²θ

So,

Cos(2θ) = Cos²θ-( 1-Cos²θ )

Cos2θ = Cos²θ - 1 + Cos²θ

Cos(2θ) = 2Cos²θ-1

LHS = RHS

Proved.

✔

Answer 2

Given :

$\cos 2\theta = 2\cos^2\theta-1$

To Find :

$\text{Show that RHS = LHS}$

Explanation :

We can prove given equations by applying some trigonometric identities.

Solution :

$\text{RHS}=\cos2\theta$

$\implies \cos (\theta+\theta)$

$\implies \cos \theta\cos\theta-\sin\theta\sin\theta$

$(\because \cos(A+B)=\cos A \cos B-\sin A \sin B)$

$\implies \cos^2A-\sin^2A$

$\implies \cos^2A-(1-\cos ^2A)$

$(\because \sin^2A=1-\cos ^2 A)$

$\implies \cos^2A-1+\cos^2A$

$\implies 2\cos^2 A-1 = \text{LHS}$

Hence Showed