Prove that:
cos²Θ - sin²Θ = 2cos²Θ - 1
You should be aware that
sin^2È + cos^2È = 1
Substitute 1 - cos^2È for sin^2È
Thank You!!!
To prove the given trigonometric identity, we can start by applying the Pythagorean identity, which states:
sin²Θ + cos²Θ = 1.
Rearranging the above equation, we get:
sin²Θ = 1 - cos²Θ.
Now, let's substitute this expression for sin²Θ in the original equation:
cos²Θ - sin²Θ = 2cos²Θ - 1.
Substituting 1 - cos²Θ for sin²Θ:
cos²Θ - (1 - cos²Θ) = 2cos²Θ - 1.
Now, distribute the negative sign:
cos²Θ - 1 + cos²Θ = 2cos²Θ - 1.
Combine like terms:
2cos²Θ - 1 = 2cos²Θ - 1.
As both sides of the equation are identical, we have proven that:
cos²Θ - sin²Θ = 2cos²Θ - 1.
Thus, the identity is true.