Question

Rezolvati ecuatia :

sin(x) + cos(x) = sin(x) * cos(x)

Answer 1

$sinx + cosx = sinx\cdot cosx|^2 \Rightarrow \\ \Rightarrow sin^2x+2\cdot sinx \cdot cosx+cos^2 x = sin^2x\cdot cos^2x \Rightarrow \\ \\ \Rightarrow 1+ sin2x = \frac{2^2\cdot sin^2x\cdot cos^2x}{4} \Rightarrow 1+sin2x = \frac{sin^22x}{4} \Rightarrow \\ \\ \Rightarrow 4+4sin2x = sin^22x\Rightarrow sin^22x-4sin2x-4=0 \\ \\ \ Notam sin2x = t, \quad t \in[-1,1] \\ \\ \RIghtarrow t^2-4t-4 = 0 \\ \\ \Delta = 16 + 16 = 32 \\ \\ t_{1,2} = \frac{4\pm4 \sqrt{2} }{2} \Rightarrow t_{1,2} = 2\pm 2 \sqrt{2}$

$t_1 = 2+2 \sqrt{2} \in [-1,1] \\ t_2 = 2-2 \sqrt{2} \notin[-1,1] \\ \\ \RIghtarrow t = 2+ \sqrt{2} \\ \\ sin2x = 2+ 2\sqrt{2} \Rightarrow 2x = \big\{(-1)^k\cdot arcsin(2+ 2\sqrt{2}) + k\pi \big\} \Rightarrow \\ \Rightarrow x = \big\{(-1)^k\cdot \frac{arcsin(2+ 2\sqrt{2}) }{2} + \frac{k\pi }{2} \big\}, \quad k \in \mathbb_{Z}$