Trigonometric Equations
Product-to-Sum — Number of Elements in Set
nta_pyq_2026_jan
Grade 11
Question:
The number of elements in the set $\{x\in[0,180°]:\tan(x+100°)=\tan(x+50°)\tan x\tan(x-50°)\}$ is _____.
Step-by-Step Solution
Key Concept: Rewrite: $\tfrac{\tan(x+100°)}{\tan x}=\tan(x+50°)\tan(x-50°)$. Apply componendo-dividendo: $\tfrac{\sin(2x+100°)}{\sin100°}=\tfrac{-\cos2x}{\cos100°}$... simplify to $\sin(4x+100°)=-\cos50°=\sin(-40°)$.
4 elements.
Correct Answer: 4