Question

The number of solutions of the equation \[4 \sin^2 x - 4 \cos^3 x + 9 - 4 \cos x = 0, \, x \in [-2\pi, 2\pi]\]is:

• A. 1
• B. 3
• C. 2
• D. 0

Answer 1

The Correct Option is D

Given equation:

\[ 4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0 \]

We use the identity:

\[ \sin^2 x = 1 - \cos^2 x \]

Substituting this in the equation:

\[ 4(1 - \cos^2 x) - 4\cos^3 x + 9 - 4\cos x = 0 \]

Simplifying:

\[ 4 - 4\cos^2 x - 4\cos^3 x + 9 - 4\cos x = 0 \]

Combining like terms:

\[ 13 - 4\cos^3 x - 4\cos^2 x - 4\cos x = 0 \]

Factoring out \(-4\):

\[ -4(\cos^3 x + \cos^2 x + \cos x - \frac{13}{4}) = 0 \]

Therefore, we need to solve:

\[ \cos^3 x + \cos^2 x + \cos x - \frac{13}{4} = 0 \]

After analyzing the roots of this equation within the interval \( x \in [-2\pi, 2\pi] \), we observe there are no real solutions that satisfy the equation.

Conclusion: The number of solutions is 0.