Question

consider the derivation of an alternate form of the cosine double angle identity.

	step
2	$= \cos^2(x) - (1 - \cos^2(x))$
3	$= \cos^2(x) - 1 - \cos^2(x)$
4	$= 2\cos^2(x) - 1$

what is the error in this derivation?

• $\bigcirc$ in step 1, $\cos(2x)$ is equal to $\cos^2(x) + \sin^2(x)$.
• $\bigcirc$ in step 2, $\sin^2(x)$ should have been replaced with $1 + \cos^2(x)$.
• $\bigcirc$ in step 3, $\cos^2(x) - 1 - \cos^2(x)$ should be $\cos^2(x) - 1 + \cos^2(x)$.
• $\bigcirc$ in step 4, $2\cos^2(x) - 1$ should be $1 - 2\cos^2(x)$.

Explanation:

To find the error, we analyze each step using the Pythagorean identity \(\sin^{2}(x)+\cos^{2}(x) = 1\), so \(\sin^{2}(x)=1 - \cos^{2}(x)\).

• Step 1: \(\cos(2x)=\cos^{2}(x)-\sin^{2}(x)\) is correct (double - angle identity).
• Step 2: To replace \(\sin^{2}(x)\), we use \(\sin^{2}(x)=1 - \cos^{2}(x)\). So \(\cos^{2}(x)-\sin^{2}(x)=\cos^{2}(x)-(1 - \cos^{2}(x))\) is correct.
• Step 3: When distributing the negative sign in \(\cos^{2}(x)-(1 - \cos^{2}(x))\), we should get \(\cos^{2}(x)-1+\cos^{2}(x)\), not \(\cos^{2}(x)-1 - \cos^{2}(x)\). The original step 3 has an error in the distribution of the negative sign.
• Step 4: If step 3 was correct, step 4 would follow, but the error is in step 3. The option about step 3 ("In step 3, \(\cos^{2}(x)-1 - \cos^{2}(x)\) should be \(\cos^{2}(x)-1+\cos^{2}(x)\)") is the correct description of the error. The other options are incorrect:
• The first option is wrong because \(\cos(2x)

eq\cos^{2}(x)+\sin^{2}(x)\) (since \(\cos^{2}(x)+\sin^{2}(x) = 1\) and \(\cos(2x)\) is not always 1).

• The second option is wrong because \(\sin^{2}(x)=1 - \cos^{2}(x)\), not \(1+\cos^{2}(x)\).
• The fourth option is wrong because \(2\cos^{2}(x)-1\) is a correct form of the double - angle identity for cosine.

Answer:

In step 3, \(\boldsymbol{\cos^{2}(x)-1 - \cos^{2}(x)}\) should be \(\boldsymbol{\cos^{2}(x)-1+\cos^{2}(x)}\). (The option corresponding to this description, e.g., if the options are labeled as above, the correct option is the one stating "In step 3, \(\cos^{2}(x)-1 - \cos^{2}(x)\) should be \(\cos^{2}(x)-1+\cos^{2}(x)\)")