3by observation x=y=pi/2
If x, y \varepsilon ( 0,\pi) and (sin 3 x + sin 4 y) + 3 cosec x + 4 cosec y - 9 =0, then no. of solutions of the equation sin x sin y = 12 in ( 0,\pi) is
A) 0 B) 1 C) 2 D) infinite
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2 Answers
341Since sin x>0 in the given domain, \sin^3 x + 3 \csc x = \sin^3x + \frac{1}{\sin x}+\frac{1}{\sin x}+\frac{1}{\sin x} \ge 4 (by AM-GM Inequality)
Similarly,
\sin^4 y + 4 \csc y \ge 5
Hence LHS ≥0 with equality when sin x = sin y = 1.
Hence sin x sin y = 1/2 has no solutions
