If `cos 5theta=a cos theta+ b cos^(3) theta+c cos^(5)theta+d`, then

Options

A. `a=20`
B. `b=-30`
C. `a+b+c=2`
D. `a+b+c+d=1`

Answer

To solve the equation \( \cos 5\theta = a \cos \theta + b \cos^3 \theta + c \cos^5 \theta + d \), we will use trigonometric identities and properties of cosine functions.

### Step 1: Use the Cosine of Sum Formula
We start by expressing \( \cos 5\theta \) in terms of \( \cos \theta \). We can use the angle addition formula:
\[ \cos 5\theta = \cos(3\theta + 2\theta) = \cos 3\theta \cos 2\theta - \sin 3\theta \sin 2\theta \]

### Step 2: Expand \( \cos 3\theta \) and \( \cos 2\theta \)
Using known identities:
\[ \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \]
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Substituting these into our equation gives:
\[ \cos 5\theta = (4\cos^3 \theta - 3\cos \theta)(2\cos^2 \theta - 1) - \sin 3\theta \sin 2\theta \]

### Step 3: Expand the Product
Expanding the product:
\[ \cos 5\theta = (4\cos^3 \theta \cdot 2\cos^2 \theta - 4\cos^3 \theta) - (3\cos \theta \cdot 2\cos^2 \theta - 3\cos \theta) \]
This simplifies to:
\[ \cos 5\theta = 8\cos^5 \theta - 4\cos^3 \theta - 6\cos^3 \theta + 3\cos \theta \]
Combining like terms:
\[ \cos 5\theta = 8\cos^5 \theta - 10\cos^3 \theta + 3\cos \theta \]

### Step 4: Identify Coefficients
Now we can compare this with the original equation:
\[ \cos 5\theta = a \cos \theta + b \cos^3 \theta + c \cos^5 \theta + d \]
From the comparison, we find:
- \( a = 3 \)
- \( b = -10 \)
- \( c = 8 \)
- \( d = 0 \)

### Step 5: Find \( a + b + c + d \)
Now, we calculate:
\[ a + b + c + d = 3 - 10 + 8 + 0 = 1 \]

### Conclusion
Thus, the values of \( a, b, c, d \) are \( 3, -10, 8, 0 \) respectively, and \( a + b + c + d = 1 \).