Question

consider the derivation of sin(2x)

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

where is the error?

• step 2 should read = cos(x)cos(x) − sin(x)sin(x)
• step 2 should read = sin(x)cos(x) − cos(x)sin(x)
• step 3 should read = −2sin(x)cos(x)
• step 3 should read = 2sin(x)cos(x)

Explanation:

Step1: Recall the sine addition formula

The formula for \(\sin(A + B)\) is \(\sin(A)\cos(B) + \cos(A)\sin(B)\). For \(A = B = x\), \(\sin(x + x)=\sin(x)\cos(x)+\cos(x)\sin(x)\), so Step 2 is correct.

Step2: Simplify \(\sin(x)\cos(x)+\cos(x)\sin(x)\)

Combining like terms, \(\sin(x)\cos(x)+\cos(x)\sin(x) = 2\sin(x)\cos(x)\). But in Step 3, it is written as \((\sin(x)\cos(x))^2\), which is wrong. The correct simplification of \(\sin(x)\cos(x)+\cos(x)\sin(x)\) is \(2\sin(x)\cos(x)\), so the error is in Step 3, and it should read \(= 2\sin(x)\cos(x)\).

Answer:

Step 3 should read \(= 2\sin(x)\cos(x)\) (the last option: "Step 3 should read \(= 2\sin(x)\cos(x)\)")