Question

which derivation correctly uses the cosine sum identity to prove the cosine double angle identity?
step 1: $cos(2x) = cos(x + x)$
step 2: $= cos(x)cos(x) - sin(x)sin(x)$
step 3: $= cos^2(x) - sin^2(x)$

step 1: $cos(2x) = cos(x + x)$
step 2: $= cos(x)cos(x) + sin(x)sin(x)$
step 3: $= cos^2(x) + sin^2(x)$

step 1: $cos(2x) = cos(x + x)$
step 2: $= sin(x)sin(x) + cos(x)cos(x)$

Explanation:

The cosine sum identity is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. To prove the cosine double - angle identity $\cos(2x)$, we let $A = B=x$. So $\cos(2x)=\cos(x + x)$. Then, by the cosine sum identity, $\cos(x + x)=\cos x\cos x-\sin x\sin x=\cos^{2}x-\sin^{2}x$.

Looking at the options:

• The first option: Step 1 writes $\cos(2x)=\cos(x + x)$, Step 2 applies the cosine sum identity correctly as $\cos(x)\cos(x)-\sin(x)\sin(x)$, and Step 3 simplifies to $\cos^{2}(x)-\sin^{2}(x)$, which is the correct cosine double - angle identity.
• The second option: In Step 2, it uses a wrong sign in the cosine sum identity (it should be minus, not plus), so it's incorrect.
• The third option: The second step has a wrong order and also the wrong sign (should be $\cos A\cos B-\sin A\sin B$, not $\sin A\sin B+\cos A\cos B$) and is incomplete, so it's incorrect.

Answer:

The first derivation (the one with the blue dot) is correct. The steps are:

1. $\cos(2x)=\cos(x + x)$ (Expressing $2x$ as a sum of two $x$ terms)
2. $=\cos(x)\cos(x)-\sin(x)\sin(x)$ (Applying the cosine sum identity $\cos(A + B)=\cos A\cos B-\sin A\sin B$ with $A = B=x$)
3. $=\cos^{2}(x)-\sin^{2}(x)$ (Simplifying the product terms)