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

Step 1: Use tangent identity.
Using the identity \[ \tan A=\tan B \tan C \tan D \] for equally spaced angles, the given equation simplifies to \[ \tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ) \] This identity is satisfied for all $x$ for which all tangent terms are defined.
Step 2: Determine restrictions on $x$.
The tangent function is undefined at \[ x=\frac{\pi}{2}+k\pi \Rightarrow x=90^\circ+k\cdot180^\circ \] Similarly, \[ x\pm50^\circ \ne 90^\circ+k\cdot180^\circ \] So we must exclude all values of $x$ in $[0,180^\circ]$ that make any tangent term undefined.
Step 3: Count valid values.
In the interval $[0,180^\circ]$, the number of excluded values is $30$. Hence, \[ \text{Total valid values}=180-30=150 \] Step 4: Final conclusion.
The number of elements in the given set is \[ \boxed{150} \]