The number of elements in the set {x ∈ [0, 180°] : tan(x + 100°) = tan(x + 50°) tan x tan(x − 50°)} is ____

The number of elements in the set {x ∈ [0, 180°] : tan(x + 100°) = tan(x + 50°) tan x tan(x − 50°)} is

Q. The number of elements in the set \[ \{x \in [0,180^\circ] : \tan(x+100^\circ) = \tan(x+50^\circ)\tan x \tan(x-50^\circ)\} \] is

Correct Answer: 4

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.