Question: Please help in Solving the Trigonometric Equation:

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

Answer: 
$$\begin{align\*}
\cos^2x-\sin^2x &= \cos3x\\
\cos2x &= \cos3x\\
2x &= 2\pi n\pm 3x\\
x &= 2\pi n \text{ or } x=\frac{2\pi n}{5}
\end{align\*}$$
where $n\in\mathbb Z$.

@ThomasAndrews: The second case includes the first case. $2\pi n=\frac{2\pi(5n)}5$.
**Hint:** Use the trigonometry identity for the $\cos 2\theta$.