work: HW 1-1.1 and 1.3 Question 24, 1.3.31 Part 3 of 6 n list $$ 11 \quad 14 $$ on 15 on 16 ion 17 ion 18 tion 19 tion 20 stion 21 stion 22 Use an addition formula to derive the identity. $$ \cos \left(x-\frac{\pi}{2}\right)=\sin x $$ Which addition formula should be used? $$ \cos (A+B) $$ $$ \boldsymbol{\operatorname { s i n }}(A+B) $$ What is the addition formula for $$ \cos (A+B) $$ ? A. $$ \cos (A+B)=\cos A \cos B-\sin A \sin B $$ B. $$ \cos (A+B)=\cos A \cos B+\sin A \sin B $$ C. $$ \cos (A+B)=\cos A \sin B+\boldsymbol{\operatorname { s i n }} A \cos B $$ D. $$ \cos (A+B)=\sin A \sin B-\cos A \cos B $$ E. $$ \cos (A+B)=\sin A \cos B-\cos A \sin B $$ F. $$ \cos (A+B)=\sin A \cos B+\cos A \sin B $$ Apply the addition formula for $$ \cos (A+B) $$ to the left side of the given identity. $$ \cos \left(x-\frac{\pi}{2}\right)=\cos (\square) \cos (\square)-\sin (\square) \sin (\square) $$ (Type the terms of your expression in the same order as they appear in the original expression. Type exact answers, using $$ \pi $$ as needed.)

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To derive the identity $$\cos \left(x-\frac{\pi}{2}\right)=\sin x$$, we will use the addition formula for cosine.

### Step 1: Identify the Addition Formula
The addition formula for cosine is:
$$
\cos (A+B) = \cos A \cos B - \sin A \sin B
$$
Thus, we will use option A.

### Step 2: Apply the Addition Formula
In our case, we can rewrite $$\cos \left(x - \frac{\pi}{2}\right)$$ as $$\cos \left(x + \left(-\frac{\pi}{2}\right)\right)$$. Here, we can identify:
- $$A = x$$
- $$B = -\frac{\pi}{2}$$

Now, we can apply the addition formula:
$$
\cos \left(x - \frac{\pi}{2}\right) = \cos(x + (-\frac{\pi}{2})) = \cos x \cos \left(-\frac{\pi}{2}\right) - \sin x \sin \left(-\frac{\pi}{2}\right)
$$

### Step 3: Evaluate $$\cos$$ and $$\sin$$ at $$-\frac{\pi}{2}$$
Using the known values:
- $$\cos \left(-\frac{\pi}{2}\right) = 0$$
- $$\sin \left(-\frac{\pi}{2}\right) = -1$$

### Step 4: Substitute the Values
Substituting these values into the formula gives:
$$
\cos \left(x - \frac{\pi}{2}\right) = \cos x \cdot 0 - \sin x \cdot (-1)
$$
This simplifies to:
$$
\cos \left(x - \frac{\pi}{2}\right) = 0 + \sin x = \sin x
$$

### Conclusion
Thus, we have derived the identity:
$$
\cos \left(x - \frac{\pi}{2}\right) = \sin x
$$

### Filling in the Blanks
Now, we can fill in the blanks for the expression:
$$
\cos \left(x - \frac{\pi}{2}\right) = \cos (x) \cos \left(-\frac{\pi}{2}\right) - \sin (x) \sin \left(-\frac{\pi}{2}\right)
$$
So, the terms of your expression in the same order as they appear in the original expression are:
- First blank: $$x$$
- Second blank: $$-\frac{\pi}{2}$$
- Third blank: $$x$$
- Fourth blank: $$-\frac{\pi}{2}$$

Thus, the final expression is:
$$
\cos \left(x - \frac{\pi}{2}\right) = \cos (x) \cos \left(-\frac{\pi}{2}\right) - \sin (x) \sin \left(-\frac{\pi}{2}\right)
$$
To derive the identity $$\cos \left(x - \frac{\pi}{2}\right) = \sin x$$, use the cosine addition formula: $$ \cos (A + B) = \cos A \cos B - \sin A \sin B $$ Apply it to $$\cos \left(x - \frac{\pi}{2}\right)$$ by setting $$A = x$$ and $$B = -\frac{\pi}{2}$$: $$ \cos \left(x - \frac{\pi}{2}\right) = \cos x \cos \left(-\frac{\pi}{2}\right) - \sin x \sin \left(-\frac{\pi}{2}\right) $$ Since $$\cos \left(-\frac{\pi}{2}\right) = 0$$ and $$\sin \left(-\frac{\pi}{2}\right) = -1$$, the expression simplifies to: $$ \cos \left(x - \frac{\pi}{2}\right) = 0 + \sin x = \sin x $$ Thus, the identity is proven.

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