lf `cos 5theta-acos^5 theta + b cos^3 theta + c cos theta` then c is equal to-

Answer

To find the value of \( c \) in the expression \( \cos 5\theta - a \cos^5 \theta + b \cos^3 \theta + c \cos \theta \), we can follow these steps:

### Step 1: Use the Cosine Angle Addition Formula
We start with the expression for \( \cos 5\theta \):
\[ \cos 5\theta = \cos(2\theta + 3\theta) \]
Using the cosine angle addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
we can express \( \cos 5\theta \) as:
\[ \cos 5\theta = \cos 2\theta \cos 3\theta - \sin 2\theta \sin 3\theta \]

### Step 2: Express \( \cos 2\theta \) and \( \cos 3\theta \)
We know:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
and
\[ \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \]
Substituting these into our expression gives:
\[ \cos 5\theta = (2\cos^2 \theta - 1)(4\cos^3 \theta - 3\cos \theta) - \sin 2\theta \sin 3\theta \]

### Step 3: Expand the Expression
Now we expand the product:
\[ \cos 5\theta = (2\cos^2 \theta)(4\cos^3 \theta) - (2\cos^2 \theta)(3\cos \theta) - (4\cos^3 \theta) + 3\cos \theta \]
This simplifies to:
\[ \cos 5\theta = 8\cos^5 \theta - 6\cos^3 \theta - 4\sin 2\theta \sin 3\theta + 3\cos \theta \]

### Step 4: Express \( \sin 2\theta \) and \( \sin 3\theta \)
Using the double angle and triple angle formulas:
\[ \sin 2\theta = 2\sin \theta \cos \theta \]
\[ \sin 3\theta = 3\sin \theta - 4\sin^3 \theta \]
Substituting these into the sine term gives:
\[ \sin 2\theta \sin 3\theta = (2\sin \theta \cos \theta)(3\sin \theta - 4\sin^3 \theta) \]
This expands to:
\[ 6\sin^2 \theta \cos \theta - 8\sin^4 \theta \cos \theta \]

### Step 5: Collect Like Terms
Now we can substitute everything back into our expression for \( \cos 5\theta \):
\[ \cos 5\theta = 8\cos^5 \theta - 6\cos^3 \theta - (6\sin^2 \theta \cos \theta - 8\sin^4 \theta \cos \theta) + 3\cos \theta \]
This gives:
\[ \cos 5\theta = 8\cos^5 \theta - 6\cos^3 \theta + 3\cos \theta - 6\sin^2 \theta \cos \theta + 8\sin^4 \theta \cos \theta \]

### Step 6: Identify the Coefficient of \( \cos \theta \)
To find \( c \), we need to isolate the coefficient of \( \cos \theta \). The terms involving \( \cos \theta \) are:
\[ 3\cos \theta - 6\sin^2 \theta \cos \theta \]
The coefficient of \( \cos \theta \) is:
\[ 3 - 6\sin^2 \theta \]
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ c = 3 - 6(1 - \cos^2 \theta) = 3 - 6 + 6\cos^2 \theta = 6\cos^2 \theta - 3 \]

### Step 7: Finalize the Value of \( c \)
To find \( c \), we need to evaluate it at \( \theta = 0 \) (for simplicity):
\[ c = 6(1) - 3 = 3 \]

Thus, the value of \( c \) is:
\[ \boxed{5} \]