anyone explain how this cos 2A cos A – sin 2A sin A
can become to this
cos (2A + A) = cos 3A
look at the definition of
cos(A+B)
cosAcosB - sinAsinB = cos(A+B)
now compare that to the given
cos2AcosA - sin2AsinA
you will reach the "inescapable" conclusion that it must be
cos(2A+A) or cos 3A
If you want the proof of why the identity is true ....
here is one by Khan
http://khanexercises.appspot.com/video/proof--cos-a-b-----cos-a--cos-b---sin-a--sin-b?playlist=Trigonometry
To understand how the expression cos 2A cos A - sin 2A sin A becomes cos (2A + A), we need to use trigonometric identities.
One of the most commonly used trigonometric identities is the cosine of the sum of angles, which states that cos (A + B) = cos A cos B - sin A sin B.
Now, let's simplify the given expression step by step:
cos 2A cos A - sin 2A sin A
Using the identity mentioned above, we can rewrite this expression as:
cos (2A + A)
Here's how we do it:
First, notice that the expression cos 2A cos A - sin 2A sin A is in the form of cos A cos B - sin A sin B, where A = 2A and B = A.
Next, substitute the values of A and B into the cosine of the sum of angles identity:
cos (2A + A) = cos 2A cos A - sin 2A sin A
As a result, cos 2A cos A - sin 2A sin A is equivalent to cos (2A + A), which further simplifies to cos 3A.