Trigonometry & Inverse Trigonometry
Inverse Trig Functions
nta_abhyas_2025
Grade 12
Question:
If $x = y = z$, $x, y, z$ are in AP, and $\tan^{-1}x$, $\tan^{-1}y$, $\tan^{-1}z$ are also in AP, find the relationship between $x$ and $z$.
Step-by-Step Solution
Key Concept: When sequences are in AP, the middle term equals the average of the endpoints; for inverse trigonometric functions in AP, use the addition formula for inverse tangent.
Since $x, y, z$ are in AP, we have $y = \frac{x+z}{2}$ (i.e., $2y = x + z$). Since $\tan^{-1}x, \tan^{-1}y, \tan^{-1}z$ are in AP, we have $2\tan^{-1}y = \tan^{-1}x + \tan^{-1}z$. This gives $\tan^{-1}\left(\frac{2y}{1-y^2}\right) = \tan^{-1}\left(\frac{x+z}{1-xz}\right)$. Substituting $2y = x+z$ and simplifying yields $\frac{2y}{1-y^2} = \frac{2y}{1-xz}$, which (for $y \neq 0$) gives $1 - y^2 = 1 - xz$, so $xz = y^2$.
Correct Answer: A