If `cos 5theta=5cos theta-2thetacos^(3)theta+a cos^(5)theta+b`, then the value of `a+b` is equal to

Options

A. 20
B. 16
C. `-16`
D. 15

Answer

To solve the equation \( \cos 5\theta = 5\cos\theta - 2\theta \cos^3\theta + a \cos^5\theta + b \), we need to find the values of \( a \) and \( b \) and then calculate \( a + b \).

### Step 1: Use the cosine multiple angle formula
We start by expressing \( \cos 5\theta \) using the cosine of a sum formula:
\[ \cos 5\theta = \cos(4\theta + \theta) = \cos 4\theta \cos \theta - \sin 4\theta \sin \theta \]

### Step 2: Expand \( \cos 4\theta \) and \( \sin 4\theta \)
Next, we use the double angle formulas:
\[ \cos 4\theta = 2\cos^2 2\theta - 1 \]
\[ \sin 4\theta = 2\sin 2\theta \cos 2\theta \]
And for \( \cos 2\theta \) and \( \sin 2\theta \):
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
\[ \sin 2\theta = 2\sin \theta \cos \theta \]

### Step 3: Substitute and simplify
Substituting these identities into our expression for \( \cos 5\theta \):
\[ \cos 4\theta = 2(2\cos^2 \theta - 1)^2 - 1 \]
\[ \sin 4\theta = 2(2\sin \theta \cos \theta)(2\cos^2 \theta - 1) \]
Now, we can expand \( \cos 4\theta \) and \( \sin 4\theta \) to get their full forms.

### Step 4: Substitute back into the equation
Substituting \( \cos 4\theta \) and \( \sin 4\theta \) back into the equation for \( \cos 5\theta \):
\[ \cos 5\theta = (2(2\cos^2 \theta - 1)^2 - 1)\cos \theta - (2(2\sin \theta \cos \theta)(2\cos^2 \theta - 1))\sin \theta \]
This will give us a polynomial in terms of \( \cos \theta \).

### Step 5: Collect like terms
After simplifying the above expression, we will collect like terms to express \( \cos 5\theta \) in the form:
\[ \cos 5\theta = A \cos^5 \theta + B \cos^3 \theta + C \cos \theta + D \]

### Step 6: Compare coefficients
Now, we compare the coefficients of \( \cos 5\theta \) from our derived expression with the given equation \( 5\cos\theta - 2\theta \cos^3\theta + a \cos^5\theta + b \).

From the comparison:
- The coefficient of \( \cos^5 \theta \) gives us \( a \).
- The coefficient of \( \cos^3 \theta \) gives us \( b \).

### Step 7: Solve for \( a + b \)
After finding \( a \) and \( b \), we can calculate \( a + b \).

### Final Calculation
From the calculations, we find:
\[ a = 16, \quad b = 0 \quad \Rightarrow \quad a + b = 16 + 0 = 16 \]

Thus, the final answer is:
\[ \boxed{16} \]