Given that `16sin^(5)x=p sin5x+qsin3x+rsinx`. What is the value of q ?

Options

A. 3
B. 5
C. 10
D. `-5

Explanation

To find the value of \( q \) in the equation \( 16 \sin^5 x = p \sin 5x + q \sin 3x + r \sin x \), we will start by expressing \( \sin 5x \) in terms of \( \sin x \) and \( \cos x \).

### Step 1: Express \( \sin 5x \)
Using the multiple angle formula for sine, we have:
\[ \sin 5x = 16 \sin^5 x - 20 \sin^3 x + 5 \sin x \]
This means we can express \( \sin^5 x \) in terms of \( \sin 5x \), \( \sin 3x \), and \( \sin x \).

### Step 2: Rewrite \( 16 \sin^5 x \)
We can rearrange the equation:
\[ 16 \sin^5 x = \sin 5x + 20 \sin^3 x - 5 \sin x \]
Now, we need to express \( \sin^3 x \) in terms of \( \sin x \) and \( \cos x \).

### Step 3: Express \( \sin^3 x \)
Using the identity \( \sin^3 x = \sin x (1 - \cos^2 x) \):
\[ \sin^3 x = \sin x (1 - (1 - \sin^2 x)) = \sin x (3 \sin^2 x - 1) \]
So, substituting back, we get:
\[ 20 \sin^3 x = 20 \sin x (3 \sin^2 x - 1) = 60 \sin^3 x - 20 \sin x \]

### Step 4: Substitute into the equation
Now substituting this back into our equation:
\[ 16 \sin^5 x = \sin 5x + 60 \sin^3 x - 20 \sin x - 5 \sin x \]
This simplifies to:
\[ 16 \sin^5 x = \sin 5x + 60 \sin^3 x - 25 \sin x \]

### Step 5: Compare coefficients
Now we can compare the coefficients of \( \sin 5x \), \( \sin 3x \), and \( \sin x \) from both sides of the equation:
- Coefficient of \( \sin 5x \) gives \( p = 1 \)
- Coefficient of \( \sin 3x \) gives \( q = -20 \)
- Coefficient of \( \sin x \) gives \( r = 10 \)

### Final Answer
Thus, the value of \( q \) is:
\[ \boxed{-20} \]