review the incomplete derivation of the cosine sum identity. which expressions for step 3 and step 5…

review the incomplete derivation of the cosine sum identity. which expressions for step 3 and step 5 complete the derivation? step 1 cos(x + y) step 2 sin((π/2)(x + y)) step 3 step 4 sin((π/2 - x)cos(y) - cos((π/2 - x)sin(y)) step 5 step 3: sin((π/2 - x) + y) step 5: cos(x)cos(y) + sin(x)sin(y) step 3: sin((π/2 - x) + y) step 5: cos(x)cos(y) - sin(x)sin(y) step 3: sin((π/2 - x) - y) step 5: cos(x)cos(y) + sin(x)sin(y) step 3: sin((π/2 - x) - y) step 5: cos(x)cos(y) - sin(x)sin(y)

review the incomplete derivation of the cosine sum identity. which expressions for step 3 and step 5 complete the derivation? step 1 cos(x + y) step 2 sin((π/2)(x + y)) step 3 step 4 sin((π/2 - x)cos(y) - cos((π/2 - x)sin(y)) step 5 step 3: sin((π/2 - x) + y) step 5: cos(x)cos(y) + sin(x)sin(y) step 3: sin((π/2 - x) + y) step 5: cos(x)cos(y) - sin(x)sin(y) step 3: sin((π/2 - x) - y) step 5: cos(x)cos(y) + sin(x)sin(y) step 3: sin((π/2 - x) - y) step 5: cos(x)cos(y) - sin(x)sin(y)

Answer

Explanation:

Step1: Recall the co - function identity

We know that $\cos(A)=\sin(\frac{\pi}{2}-A)$. So, $\cos(x + y)=\sin(\frac{\pi}{2}-(x + y))=\sin((\frac{\pi}{2}-x)-y)$.

Step2: Use the sine - difference formula

The sine - difference formula is $\sin(A - B)=\sin(A)\cos(B)-\cos(A)\sin(B)$. For $\sin((\frac{\pi}{2}-x)-y)$, we have $\sin((\frac{\pi}{2}-x)-y)=\sin(\frac{\pi}{2}-x)\cos(y)-\cos(\frac{\pi}{2}-x)\sin(y)$. We know that $\sin(\frac{\pi}{2}-x)=\cos(x)$ and $\cos(\frac{\pi}{2}-x)=\sin(x)$. So, $\sin(\frac{\pi}{2}-x)\cos(y)-\cos(\frac{\pi}{2}-x)\sin(y)=\cos(x)\cos(y)-\sin(x)\sin(y)$.

Answer:

Step 3: $\sin((\frac{\pi}{2}-x)-y)$ Step 5: $\cos(x)\cos(y)-\sin(x)\sin(y)$