The value of ` cos ( 35^(@) + A)cos ( 35^(@) - B) + sin ( 35^(@) +A)sin ( 35^(@) - B)` is equal to (i) `sin(A+ B) ` (ii) `sin(A - B)` (iii) `cos (A+B)` (iv) `cos(A-B)`

Options

A. `sin(A+ B) `
B. `sin(A - B)`
C. `cos (A+B)`
D. `cos(A-B)`

Answer

To solve the problem, we need to simplify the expression given:

\[ \cos(35^\circ + A) \cos(35^\circ - B) + \sin(35^\circ + A) \sin(35^\circ - B) \]

### Step 1: Identify the Trigonometric Identity
We can use the cosine of the sum and difference identity, which states that:

\[ \cos(x - y) = \cos x \cos y + \sin x \sin y \]

### Step 2: Apply the Identity
In our case, we can set \( x = 35^\circ + A \) and \( y = 35^\circ - B \). Thus, we can rewrite the expression as:

\[ \cos((35^\circ + A) - (35^\circ - B)) \]

### Step 3: Simplify the Argument
Now, simplify the argument of the cosine:

\[ (35^\circ + A) - (35^\circ - B) = 35^\circ + A - 35^\circ + B = A + B \]

### Step 4: Write the Final Result
So, we have:

\[ \cos(35^\circ + A) \cos(35^\circ - B) + \sin(35^\circ + A) \sin(35^\circ - B) = \cos(A + B) \]

### Conclusion
Thus, the value of the expression is:

\[ \cos(A + B) \]

### Step 5: Identify the Correct Option
From the options provided, the correct answer is:

(iii) \( \cos(A + B) \)

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