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Using trig identities

Q:  Find cos(2a) if we know that (a – 2 cos a) = 0.

A:  If (a – 2cos a) = 0, we can solve for “cos(a)” like so:

a – 2cos a = 0

-2cos a = -a

cos a = -a / -2

cos a = a / 2

Now, we must recall one of our trig identities — a double angle identity to be exact.  Why do I know to do this?  Because I know we are trying to find cos (2a)… and that looks like a double angle identity.  I have a trig identity that tells me:

cos(2x) = 2cos²x – 1

Therefore,

cos(2a) = 2cos²a – 1

Since cos a = a/2, we can subtitute like so:

cos(2a) = 2(a/2)² – 1

cos(2a) = 2(a²/2²) – 1

cos(2a) = 2(a²/4) – 1

cos(2a) = 2a²/4 – 1

cos(2a) = a²/2 – 1

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