How do i solve for this identity.....
cos2theta=2-sec^2theta/sec^2theta
To solve for the given identity cos(2θ) = (2 - sec^2θ) / sec^2θ, we can use some trigonometric identities to simplify the equation.
First, let's rewrite sec^2θ as 1/cos^2θ:
cos(2θ) = (2 - 1/cos^2θ) / (1/cos^2θ)
Next, let's get rid of the fraction by multiplying both sides of the equation by cos^2θ:
cos^2(2θ) = 2 - 1/cos^2θ
Now, we can simplify the equation by expanding cos^2(2θ):
1 - sin^2(2θ) = 2 - 1/cos^2θ
Using the double-angle identity for sine, sin^2(2θ) = (1 - cos(4θ))/2, we can substitute this into the equation:
1 - (1 - cos(4θ))/2 = 2 - 1/cos^2θ
Next, let's simplify the equation further by multiplying through by 2:
2 - (1 - cos(4θ)) = 4 - 2/cos^2θ
Simplifying the equation gives:
2 - 1 + cos(4θ) = 4 - 2/cos^2θ
Now, we can simplify further:
cos(4θ) = 3 - 2/cos^2θ
Finally, solving for cos(4θ), we rearrange the equation:
cos(4θ) = 3 - 2/cos^2θ
This is the simplified form of the identity cos(2θ) = (2 - sec^2θ) / sec^2θ.