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

Explanation:

Step1: Use the co - function identity

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

Step2: Use the sine difference formula

The sine difference formula is (\sin(A - B)=\sin(A)\cos(B)-\cos(A)\sin(B)). Here (A=\frac{\pi}{2}-x) and (B = y), so (\sin\left(\left(\frac{\pi}{2}-x\right)-y\right)=\sin\left(\frac{\pi}{2}-x\right)\cos(y)-\cos\left(\frac{\pi}{2}-x\right)\sin(y))

Step3: Use co - function identities again

We know that (\sin\left(\frac{\pi}{2}-x\right)=\cos(x)) and (\cos\left(\frac{\pi}{2}-x\right)=\sin(x)) [ \begin{align*} \sin\left(\frac{\pi}{2}-x\right)\cos(y)-\cos\left(\frac{\pi}{2}-x\right)\sin(y)&=\cos(x)\cos(y)-\sin(x)\sin(y) \end{align*} ]

Answer:

Step 3: (\sin\left(\left(\frac{\pi}{2}-x\right)-y\right)), Step 5: (\cos(x)\cos(y)-\sin(x)\sin(y)) (the fourth option)