Inverse Trigonometric Functions

Let

Step 1: Expressing in Terms of Half-Angle

Therefore,

Step 2: Calculating Individual Trigonometric Functions

Step 3: Applying the Tangent Subtraction Formula

Final Calculation

Step 1: Let . Then Substitute in the given equation:
Step 2: Apply and the domain restriction We rewrite the equation:
Step 3: Define LHS = RHS and analyze the function: Let We want to solve
Step 4: Observe that So, Step 5: Let us analyze So define a function: We now find the number of real solutions of
Step 6: Graphical/Monotonicity Analysis (or plotting): Using symmetry and bounds: - -
Graphing shows that it crosses the x-axis three times. So, the number of real solutions is 3.