The number of solutions of the equation $\cos 2\theta\cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$ in $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ is:

Step 1: Rearrange the equation

The given equation is $\cos 2\theta\cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$. We can rearrange this equation by moving the term $\cos\frac{5\theta}{2}$ to the right side.

Step 2: Factor the right side using the triple angle identity

We can factor out $\cos\frac{5\theta}{2}$ from the right side. The expression $2\cos^2 x - 1$ is the double angle identity for $\cos 2x$. Here, $x = \frac{5\theta}{2}$, so $2\cos^2\frac{5\theta}{2} - 1 = \cos(2 \cdot \frac{5\theta}{2}) = \cos 5\theta$.

Step 3: Apply product-to-sum identities

We use the product-to-sum identity $2\cos A \cos B = \cos(A+B) + \cos(A-B)$ on both sides of the equation. This transforms the products of cosines into sums of cosines.

Step 4: Simplify the arguments and solve

After applying the product-to-sum identities and simplifying the arguments, we get $\cos\frac{5\theta}{2} + \cos\frac{3\theta}{2} = \cos\frac{15\theta}{2} + \cos\frac{5\theta}{2}$. The term $\cos\frac{5\theta}{2}$ cancels out from both sides, leaving $\cos\frac{3\theta}{2} = \cos\frac{15\theta}{2}$. The general solution for $\cos A = \cos B$ is $A = 2n\pi \pm B$, where $n$ is an integer.

Step 5: Solve for $\theta$ in two cases

We consider two cases based on the general solution. In Case 1, we take the positive sign, leading to $\theta = \frac{n\pi}{3}$. In Case 2, we take the negative sign, leading to $\theta = \frac{2n\pi}{9}$.

Step 6: Find solutions in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$

We need to find the values of $\theta$ that lie in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which is approximately $[-1.57, 1.57]$. For $\theta = \frac{n\pi}{3}$: $n=0, \pm 1$ give $0, \pm \frac{\pi}{3}$. For $\theta = \frac{2n\pi}{9}$: $n=0, \pm 1, \pm 2$ give $0, \pm \frac{2\pi}{9}, \pm \frac{4\pi}{9}$. The unique solutions are $0, \pm \frac{\pi}{3}, \pm \frac{2\pi}{9}, \pm \frac{4\pi}{9}$.

Step 7: Count the number of unique solutions

Listing all the unique solutions found: $0, \frac{\pi}{3}, -\frac{\pi}{3}, \frac{2\pi}{9}, -\frac{2\pi}{9}, \frac{4\pi}{9}, -\frac{4\pi}{9}$. There are 7 unique solutions.

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