consider the derivation of an alternate form of the cosine double - angle identity. what is the error in…

consider the derivation of an alternate form of the cosine double - angle identity. what is the error in this derivation? step 1. cos(2x)=cos²(x) - sin²(x) 2. =cos²(x)-(1 - cos²(x)) 3. =cos²(x)-1 - cos²(x) 4. =2cos²(x)-1 in step 1, cos(2x) is equal to cos²(x)+sin²(x). in step 2, sin²(x) should have been replaced with 1 + cos²(x). in step 3, cos²(x)-1 - cos²(x) should be cos²(x)-1 + cos²(x). in step 4, 2cos²(x)-1 should be 1 - 2cos²(x).

consider the derivation of an alternate form of the cosine double - angle identity. what is the error in this derivation? step 1. cos(2x)=cos²(x) - sin²(x) 2. =cos²(x)-(1 - cos²(x)) 3. =cos²(x)-1 - cos²(x) 4. =2cos²(x)-1 in step 1, cos(2x) is equal to cos²(x)+sin²(x). in step 2, sin²(x) should have been replaced with 1 + cos²(x). in step 3, cos²(x)-1 - cos²(x) should be cos²(x)-1 + cos²(x). in step 4, 2cos²(x)-1 should be 1 - 2cos²(x).

Answer

Explanation:

Step1: Analyze step 1

The double - angle formula $\cos(2x)=\cos^{2}(x)-\sin^{2}(x)$ is correct.

Step2: Analyze step 2

Using the Pythagorean identity $\sin^{2}(x)=1 - \cos^{2}(x)$ to substitute for $\sin^{2}(x)$ in step 1 is correct.

Step3: Analyze step 3

When expanding $\cos^{2}(x)-(1 - \cos^{2}(x))$, we should get $\cos^{2}(x)-1+\cos^{2}(x)$ (using the distributive property $a-(b - c)=a - b + c$), not $\cos^{2}(x)-1-\cos^{2}(x)$. So the error is in step 3.

Step4: Analyze step 4

Since step 3 is wrong, step 4 is also wrong as it is based on step 3.

Answer:

In step 3, $\cos^{2}(x)-1-\cos^{2}(x)$ should be $\cos^{2}(x)-1+\cos^{2}(x)$.