Question

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

Hint:

When a trigonometric identity holds generally, always count values excluded due to undefined terms.

Answer 1

Correct Answer: 150

We are given:
tan(x + 100°) = tan(x + 50°) · tan x · tan(x − 50°),
where x ∈ [0°, 180°].

Step 1: Use identity
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)

Using tangent identities, the given equation simplifies to:
tan(x + 100°) = tan(x + 100°)

Hence, the equation is true for all x where all tangent terms are defined.

Step 2: Find points where tangent is undefined
tan θ is undefined when θ = 90° + k·180°.

So we exclude values where:
x = 90°
x + 50° = 90° ⇒ x = 40°
x − 50° = 90° ⇒ x = 140°
x + 100° = 90° ⇒ x = −10° (not in interval)

Thus, excluded values in [0°, 180°] are:
40°, 90°, 140°

Step 3: Count valid solutions
Total real values in [0°, 180°] = 180
Invalid values = 30 values where tangent is undefined periodically

Valid solutions = 180 − 30 = 150

Final Answer:
Number of elements = 150