sin2 =2tan cos

To prove the equation sin^2θ = 2tanθsinθ, we'll start by using basic trigonometric identities.

First, let's rewrite tanθ as sinθ/cosθ using the definition of tangent.

So, tanθ = sinθ/cosθ.

Now, let's rearrange the equation to isolate sinθ on one side:

tanθ = sinθ/cosθ

Multiply both sides of the equation by cosθ to eliminate the denominator:

cosθ(tanθ) = sinθ

Next, we'll use the identity sin2θ = 2sinθcosθ.

Therefore, we can rewrite the left side of the equation as sin2θ:

sin2θ = 2sinθcosθ

Now, let's substitute the value of cosθ(tanθ) we found earlier:

sin2θ = cosθ(tanθ)

Since we obtained the expression cosθ(tanθ) on both sides of the equation, we can replace it with sinθ, as we showed earlier:

sin2θ = 2tanθsinθ.

Therefore, we have proven that sin^2θ = 2tanθsinθ.