simplify (cos2Ɵ-1)/sin2Ɵ
Recall your half-angle formula for tan
sin ( 2 θ ) = 2 sin θ ∙ cos θ
cos 2 θ = cos² θ - sin² θ
cos 2 θ = cos² θ - ( 1 - cos² θ )
cos 2 θ = cos² θ - 1 + cos² θ
cos 2 θ = 2 cos² θ - 1
cos 2 θ - 1 = 2 cos² θ - 1 - 1
cos 2 θ - 1 = 2 cos² θ - 2
cos 2 θ - 1 = 2 ( cos² θ - 1)
cos 2 θ - 1 = 2 ( 1 - sin² θ - 1)
cos 2 θ - 1 = - 2 sin² θ
cos 2 θ - 1 = - 2 sin θ ∙ sin θ
( cos 2 θ - 1 ) / sin 2 θ = - 2 sin θ ∙ sin θ / 2 sin θ ∙ cos θ = - sin θ / cos θ = - tan θ
( cos 2 θ - 1 ) / sin 2 θ = - tan θ
To simplify the expression (cos^2(θ) - 1) / sin^2(θ), we can use some trigonometric identities.
Let's start by expanding the numerator using the identity cos^2(θ) = 1 - sin^2(θ):
(cos^2(θ) - 1) = (1 - sin^2(θ) - 1) = -sin^2(θ)
Now, the expression becomes -sin^2(θ) / sin^2(θ).
Since sin^2(θ) appears in both the numerator and the denominator, we can cancel them out:
-sin^2(θ) / sin^2(θ) = -1
Therefore, the simplified form of (cos^2(θ) - 1) / sin^2(θ) is simply -1.