Single Choice Questions: 1.Let `0<=alpha, beta, gamma, delta,<=pi` where `beta` and `gamma` are not complementary such that `2cos alpha+ 6 cos beta+7 cos gamma+9 cos delta=0` and `2 sin alpha- 6 sin beta+7 sin gamma-9 sin delta=0` if`cos(alpha+delta)/cos(beta+gamma)=m/n`when m and n are relatively prime positive numbers, then the value of `(m+n)` is equal to (A)11 (B)10 (C)9 (D)7

Options

A. 11
B. 10
C. 9
D. 2

Answer

To solve the given problem step-by-step, we will follow the instructions outlined in the video transcript.

### Step 1: Write down the equations
We start with the two equations provided in the problem:
1. \( 2 \cos \alpha + 6 \cos \beta + 7 \cos \gamma + 9 \cos \delta = 0 \)
2. \( 2 \sin \alpha - 6 \sin \beta + 7 \sin \gamma - 9 \sin \delta = 0 \)

### Step 2: Rearranging the equations
Rearranging both equations gives us:
- From the first equation:
\[ 2 \cos \alpha + 9 \cos \delta = -6 \cos \beta - 7 \cos \gamma \]
- From the second equation:
\[ 2 \sin \alpha - 9 \sin \delta = 6 \sin \beta - 7 \sin \gamma \]

### Step 3: Square both equations and add them
Now we square both equations and add them:
\[ (2 \cos \alpha + 9 \cos \delta)^2 + (2 \sin \alpha - 9 \sin \delta)^2 = (-6 \cos \beta - 7 \cos \gamma)^2 + (6 \sin \beta - 7 \sin \gamma)^2 \]

### Step 4: Expand and simplify
Expanding both sides, we get:
- Left Side:
\[ 4 \cos^2 \alpha + 81 \cos^2 \delta + 36 \cos \alpha \cos \delta - 36 \sin \alpha \sin \delta \]
- Right Side:
\[ 36 \cos^2 \beta + 49 \cos^2 \gamma + 84 \cos \beta \cos \gamma + 36 \sin^2 \beta + 49 \sin^2 \gamma - 84 \sin \beta \sin \gamma \]

### Step 5: Use Pythagorean identity
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
- Left Side simplifies to:
\[ 4 + 81 + 36 \cos(\alpha + \delta) = 85 + 36 \cos(\alpha + \delta) \]
- Right Side simplifies to:
\[ 36 + 49 + 84 \cos(\beta + \gamma) = 85 + 84 \cos(\beta + \gamma) \]

### Step 6: Set the equations equal
Setting the two sides equal gives:
\[ 85 + 36 \cos(\alpha + \delta) = 85 + 84 \cos(\beta + \gamma) \]
This simplifies to:
\[ 36 \cos(\alpha + \delta) = 84 \cos(\beta + \gamma) \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ \frac{\cos(\alpha + \delta)}{\cos(\beta + \gamma)} = \frac{84}{36} = \frac{7}{3} \]

### Step 8: Identify m and n
From the equation, we have:
\[ \frac{\cos(\alpha + \delta)}{\cos(\beta + \gamma)} = \frac{m}{n} \quad \text{where } m = 7 \text{ and } n = 3 \]

### Step 9: Find m + n
Now, we find \( m + n \):
\[ m + n = 7 + 3 = 10 \]

### Final Answer
Thus, the value of \( m + n \) is \( \boxed{10} \).

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