Question

Use an angle sum identity to verify the identity cos 2 theta = 2 cos^2 theta - 1

Answer 1

what is angle sum identity...I can figure out which one it is on this list

Answer 2

So, we need to verify:
\[cos(2\theta)=2cos^2(\theta)-1\]
A angle sum formula is one which allows us to calculate angles in the form of \(\theta=a+b\) Where \(a\) and \(b\) are portions of the angle. For \(\{cos\}\), we have:
\[cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\]
So, we can say that \(cos(2\theta)=cos(\theta+\theta)\). Let \(a=\theta\), and \(b=\theta\); we can derive the following equation:
\(cos(\theta+\theta)=cos(2\theta)\)
And
\(cos(\theta+\theta)=cos(\theta)cos(\theta)-sin(\theta)sin(\theta)=cos^2(\theta)-sin^2(\theta)\)

So therefore, \(cos(2\theta)=cos^2(\theta)-sin^2(\theta)\). We can simplify this a little further by remembering that \(cos^2(\theta)+sin^2(\theta)=1\) and observing that \(sin^2(\theta)=1-cos^2(\theta)\)

Therefore, we can simplify the equation further to:
\[\eqalign{cos(2\theta)&=cos^2(\theta)-sin^2(\theta) \\
&=cos^2(\theta)-[1-cos^2(\theta)] \\
&=cos^2(\theta)-1+cos^2(\theta) \\
&=2cos^2(\theta)-1 \\}\]