Question

Find the number of solutions of the equation \[ \tan(x+100^\circ)=\tan(x+50^\circ)\tan(x-50^\circ) \] where $x\in(0,\pi)$.

Hint:

When solving trigonometric equations, always check final values lie in the given interval.

Answer 1

Correct Answer: 4

Step 1: Convert tangent into sine and cosine.
\[ \frac{\sin(x+100^\circ)}{\cos(x+100^\circ)} =\frac{\sin(x+50^\circ)\sin(x-50^\circ)} {\cos(x+50^\circ)\cos(x-50^\circ)} \] Step 2: Cross multiply.
\[ \sin(x+100^\circ)\cos(x+50^\circ)\cos(x-50^\circ) \] \[ =\cos(x+100^\circ)\sin(x+50^\circ)\sin(x-50^\circ) \] Step 3: Apply trigonometric identities.
\[ \sin(4x+100^\circ)=-\sin50^\circ \] Step 4: Solve general solution.
\[ 4x+100^\circ=n\pi+(-1)^n(-50^\circ) \] Step 5: Find valid $x\in(0,\pi)$.
\[ x=\frac{130^\circ}{4},\frac{210^\circ}{4},\frac{490^\circ}{4},\frac{570^\circ}{4} \] Step 6: Count solutions.
Total solutions $=4$.
Final conclusion.
The number of solutions is 4.