# If $\alpha\cos^23\theta+\beta\cos^4\theta=16\cos^6\theta+9\cos^2\theta$ is an identity then

$\begin{array}{1 1}(a)\;\alpha=1,\beta=18&(b)\;\alpha=1,\beta=24\\(c)\;\alpha=3,\beta=24&(d)\;\alpha=4,\beta=2\end{array}$

From the given identity we have
$\alpha[4\cos^3\theta-3\cos\theta]^2+\beta\cos^4\theta=16\cos^6\theta+9\cos^2\theta$
$\Rightarrow 16\alpha\cos^6\theta+(\beta-24\alpha)\cos^4\theta+9\alpha\cos^2\theta=16\cos^6\theta+9\cos^2\theta$
$\Rightarrow \alpha=1$ and $\beta-24\alpha=0$
$\Rightarrow \alpha=1$
$\beta-24\alpha=0$
$\beta=24$
$\alpha=1$ and $\beta=24$
Hence (b) is the correct option.