a student uses the following steps to prove the sine sum identity. the proof is incorrect.\nstep 1 sin(x +…

a student uses the following steps to prove the sine sum identity. the proof is incorrect.\nstep 1 sin(x + y)\nstep 2 cos(\\(\\frac{\\pi}{2}-(x + y)\\))\nstep 3 cos(\\(\\left(\\frac{\\pi}{2}-x\\right)-y\\))\nstep 4 cos(\\(\\frac{\\pi}{2}-x\\))cos(-y)+sin(\\(\\frac{\\pi}{2}-x\\))sin(-y)\nstep 5 sin(x)cos(y)+cos(x)sin(y)\nwhich step is incorrect?\nstep 2\nstep 3\nstep 4\nstep 5

a student uses the following steps to prove the sine sum identity. the proof is incorrect.\nstep 1 sin(x + y)\nstep 2 cos(\\(\\frac{\\pi}{2}-(x + y)\\))\nstep 3 cos(\\(\\left(\\frac{\\pi}{2}-x\\right)-y\\))\nstep 4 cos(\\(\\frac{\\pi}{2}-x\\))cos(-y)+sin(\\(\\frac{\\pi}{2}-x\\))sin(-y)\nstep 5 sin(x)cos(y)+cos(x)sin(y)\nwhich step is incorrect?\nstep 2\nstep 3\nstep 4\nstep 5

Answer

Explanation:

Step1: Analyze Step 2

Using the co - function identity (\sin\alpha=\cos(\frac{\pi}{2}-\alpha)), when (\alpha = x + y), (\sin(x + y)=\cos(\frac{\pi}{2}-(x + y))). So Step 2 is correct.

Step2: Analyze Step 3

Using the associative property of addition (\frac{\pi}{2}-(x + y)=(\frac{\pi}{2}-x)-y). So Step 3 is correct.

Step3: Analyze Step 4

Using the cosine of a difference formula (\cos(A - B)=\cos A\cos B+\sin A\sin B), where (A=\frac{\pi}{2}-x) and (B = y), we have (\cos((\frac{\pi}{2}-x)-y)=\cos(\frac{\pi}{2}-x)\cos y+\sin(\frac{\pi}{2}-x)\sin y). Since (\cos(-y)=\cos y) and (\sin(-y)=-\sin y), the correct expansion of (\cos((\frac{\pi}{2}-x)-y)) should be (\cos(\frac{\pi}{2}-x)\cos y+\sin(\frac{\pi}{2}-x)\sin y), not (\cos(\frac{\pi}{2}-x)\cos(-y)+\sin(\frac{\pi}{2}-x)\sin(-y)). So Step 4 is incorrect.

Step4: Analyze Step 5 (for completeness)

If Step 4 was correct, using (\cos(\frac{\pi}{2}-x)=\sin x), (\sin(\frac{\pi}{2}-x)=\cos x), (\cos(-y)=\cos y) and (\sin(-y)=-\sin y) (but in the wrong - expanded Step 4), we would have an error. But since we already found the error in Step 4, if we assume correct trigonometric identities: (\cos(A - B)=\cos A\cos B+\sin A\sin B), (A=\frac{\pi}{2}-x), (B = y), (\cos(\frac{\pi}{2}-x)=\sin x), (\sin(\frac{\pi}{2}-x)=\cos x), the correct expansion (\cos((\frac{\pi}{2}-x)-y)=\sin x\cos y+\cos x\sin y). But the error is in Step 4.

Answer:

Step 4