Question

If tan^-1x + tan^-1y = $\frac{4\mathrm{\pi }}{5}$, then cot^-1x + cot^-1y = _________________.

Answer 1

We know

${\tan }^{-1}a+{\cot }^{-1}a=\frac{\mathrm{\pi }}{2}$, for all a ∈ R .....(1)

Now,

${\tan }^{-1}x+{\tan }^{-1}y=\frac{4\mathrm{\pi }}{5}$ (Given)

$\Rightarrow \frac{\mathrm{\pi }}{2}-{\cot }^{-1}x+\frac{\mathrm{\pi }}{2}-{\cot }^{-1}y=\frac{4\mathrm{\pi }}{5}$ [Using (1)]

$\Rightarrow {\cot }^{-1}x+{\cot }^{-1}y=\mathrm{\pi }-\frac{4\mathrm{\pi }}{5}$

$\Rightarrow {\cot }^{-1}x+{\cot }^{-1}y=\frac{\mathrm{\pi }}{5}$

If tan⁻¹x + tan⁻¹y = $\frac{4\mathrm{\pi }}{5}$, then cot⁻¹x + cot⁻¹y = $\underline{\frac{\mathrm{\pi }}{5}}$.