prove cos2a = 1/2(1+cos2A) ?

Is cos2a the same as cos2A ?

if yes,

cos2a = 1/2(1+cos2A)

Multiply both sides by 2,

2cos2a = 1 +cos2A

2cos2a -cos2A = 1

cos2a = 1 ----------------No, that is not true.

That means cos2a is not the same as cos2A.

I think your cos2a should have been cos^2(A), or (cosA)^2.

That is the same as your "cos(squared)A"

Let us see,

cos^2(A) = (1/2)[1 +cos(2A)]

cos^2(A) = (1/2)[{sin^2(A) +cos^2(A)} +{cos^2(A) -sin^2(A)}]

cos^2(A) = (1/2)[sin^2(A) +cos^2(A) +cos^2(A) -sin^2(A)]

cos^2(A) = (1/2)[2cos^2(A)]

cos^2(A) = cos^2(A) -----------***

Proven.

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We used also the trig identity

sin^2(A) +cos^2(A) = 1