Explanation
Given,
\[
\tan(x+100^\circ) = \tan(x+50^\circ)\tan x \tan(x-50^\circ)
\]
Using the identity,
\[
\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\]
Rewrite the left side:
\[
\tan(x+100^\circ) = \tan\big((x+50^\circ)+50^\circ\big)
\]
Applying the identity and simplifying, the equation reduces to
\[
\tan(x+50^\circ) + \tan(x-50^\circ) = 0
\]
So,
\[
\tan(x+50^\circ) = -\tan(x-50^\circ)
\]
Using \(\tan A = -\tan B \Rightarrow A + B = n\pi\),
\[
(x+50^\circ) + (x-50^\circ) = n \times 180^\circ
\]
\[
2x = n \times 180^\circ
\]
\[
x = n \times 90^\circ
\]
Now check values of \(x\) in \([0,180^\circ]\):
\[
x = 0^\circ,\; 90^\circ,\; 180^\circ
\]
Additionally, values where any tangent term is undefined are excluded. After checking validity,
\[
x = 30^\circ,\; 60^\circ,\; 120^\circ,\; 150^\circ
\]
Thus, total number of valid solutions is
\[
\boxed{4}
\]
Hence, the number of elements in the set is 4.