Question: Please help in Solving the Trigonometric Equation:

$$\cos^2x - \sin^2x = \cos3x$$

Answer: 
$$
\cos 3x=\cos x \cos x - \sin x \sin x= \cos 2x,
$$
which is equivalent to
$$
3x\pm 2x = 2k\pi
$$
for some integer $k$.
As others have mentioned you should be after the roots of $\cos(3x)-\cos(2x)=0$. To continue from here you may use the identity $\cos \theta-\cos \alpha=-2\sin\frac{\theta-\alpha}{2}\sin\frac{\theta+\alpha}{2}$, i.e. 

$$\cos(3x)-\cos(2x)=-2\sin\frac{5x}{2}\sin\frac{x}{2}$$