use an apropriate sum or difference indentity to prove the double-angle identity cos2u=1-2sin2u
You stated the identity incorrectly , is should have been
cos 2u = 1 - 2sin^2 u
LS = cos 2u
= cos(u+u) = cosucosu - sinusinu
= cos^2 u - sin^2 u
= (1-sin^2 u) - sin^2 u
= 1 - 2sin^2 u
= RS
To prove the double-angle identity cos(2u) = 1 - 2sin^2(u), we can start with the sum or difference identity for cosine. The sum or difference identity states:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Now, let's use the double-angle identity where A = u and B = u:
cos(2u) = cos(u + u)
Using the sum identity:
cos(2u) = cos(u)cos(u) - sin(u)sin(u)
Since cos(u)cos(u) is equal to cos^2(u) and sin(u)sin(u) is equal to sin^2(u), we can rewrite the equation as:
cos(2u) = cos^2(u) - sin^2(u)
Recall the identity: cos^2(u) + sin^2(u) = 1
Rearrange the identity: cos^2(u) = 1 - sin^2(u)
Substitute this back into the previous equation:
cos(2u) = 1 - sin^2(u)
And there you have it, the double-angle identity cos(2u) = 1 - 2sin^2(u) is proven using the appropriate sum or difference identity.