How do you derive this identity?
Cos(t)=1-sin^2(t/2)
To derive the identity Cos(t) = 1 - Sin^2(t/2), we can start with the double angle formula for sine:
Sin(2θ) = 2Sin(θ)Cos(θ)
In this case, let θ = t/2:
Sin(t) = 2Sin(t/2)Cos(t/2)
Now, square both sides of the equation:
Sin^2(t) = (2Sin(t/2)Cos(t/2))^2
Using the identity for cosine squared, Cos^2(θ) = 1 - Sin^2(θ), we substitute Cos^2(t/2) with 1 - Sin^2(t/2):
Sin^2(t) = (2Sin(t/2)(1 - Sin^2(t/2)))^2
Expanding and simplifying:
Sin^2(t) = 4Sin^2(t/2)(1 - Sin^2(t/2))
Divide both sides of the equation by Sin^2(t/2):
Sin^2(t) / Sin^2(t/2) = 4(1 - Sin^2(t/2))
Cancel out Sin^2(t/2):
Sin^2(t) = 4(1 - Sin^2(t/2))
Rearrange the terms to isolate Sin^2(t/2):
Sin^2(t/2) = 1 - (1/4)Sin^2(t)
Now, we can substitute Sin^2(t/2) in the double angle formula for cosine, which is:
Cos(2θ) = Cos^2(θ) - Sin^2(θ)
In this case, let θ = t/2:
Cos(t) = Cos^2(t/2) - Sin^2(t/2)
And since Sin^2(t/2) = 1 - (1/4)Sin^2(t), we can substitute it in:
Cos(t) = Cos^2(t/2) - (1 - (1/4)Sin^2(t))
Expand and simplify:
Cos(t) = Cos^2(t/2) - 1 + (1/4)Sin^2(t)
Rearrange the terms:
Cos(t) = 1 - (1/4)Sin^2(t) + Cos^2(t/2) - 1
Cancel out the ones:
Cos(t) = 1 - (1/4)Sin^2(t) + Cos^2(t/2) - 1
Combine the terms:
Cos(t) = 1 - Sin^2(t/2) + Cos^2(t/2)
And finally, simplify:
Cos(t) = 1 - Sin^2(t/2)