If 5sin⁡xcos⁡y=1,4tan⁡ x=tan⁡ y, then - Infinity Learn by Sri Chaitanya

A
$x = (m + n) \frac{π}{2} + \frac{π}{4} + (- 1)^{n} \frac{1}{2} \sin^{- 1} (- \frac{3}{5}), m, n \in Z$
B
$y = (n - m) \frac{π}{2} + \frac{π}{4} + (- 1)^{m + 1} \frac{1}{2} \sin^{- 1} (- \frac{3}{5}); m, n \in Z$
C
$x = (m + n) \frac{π}{2} + \frac{π}{4} + (- 1)^{m} \frac{1}{2} \sin^{- 1} (\frac{3}{5}); m, n \in Z$
D
$y = (n - m) \frac{π}{2} + \frac{π}{4} + (- 1)^{m + 1} \frac{1}{2} \sin^{- 1} (\frac{3}{5}); m, n \in Z$

We have $5 \sin x \cos y = 1$ and $4 \tan x = \tan y$

$∴ 5 \sin x \cos y = 1$ and $4 \sin x \cos y = \sin y \cos x$

$\Rightarrow \sin x \cos y = \frac{1}{5}$ and $\cos x \sin y = \frac{4}{5}$

$\Rightarrow \sin x \cos y + \cos x \sin y = 1$

$\Rightarrow \sin (x + y) = \sin \frac{π}{2}$

and $\sin x \cos y - \cos x \sin y = - \frac{3}{5}$

$\Rightarrow \sin (x - y) = \sin \{\sin^{- 1} (- \frac{3}{5})\}$

$\Rightarrow x + y = (2 n + 1) \frac{π}{2}, n \in Z$

and $x - y = m π + (- 1)^{m} \sin^{- 1} (- \frac{3}{5}); m \in Z$

$\Rightarrow x = (m + n) \frac{π}{2} + \frac{π}{4} + (- 1)^{m} \frac{1}{2} \sin^{- 1} (- \frac{3}{5}); m, n \in Z$

and $y = (n - m) \frac{π}{2} + \frac{π}{4} + (- 1)^{m + 1} \frac{1}{2} \sin^{- 1} (- \frac{3}{5}); m, n \in Z$

If $5 \sin x \cos y = 1, 4 \tan x = \tan y$ , then