Trigonometry & Inverse Trigonometry
Equation Involving arctan — Number of Solutions
nta_pyq_2024_jan
Grade 12
Question:
Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x)+\tan^{-1}(2x)=\dfrac{\pi}{4}$ is:
Step-by-Step Solution
Key Concept: Apply $\tan$ to both sides after rearranging: $\tan^{-1}(2x)=\pi/4-\tan^{-1}(x)$. Take $\tan$ of both sides: $2x=\frac{1-x}{1+x}$. Solve the resulting quadratic and check which roots are positive and satisfy the principal value constraints.
$2x^2+3x-1=0\Rightarrow x=\frac{-3+\sqrt{17}}{8}$ (positive root). One positive solution.
Correct Answer: 2