Răspuns:
[tex]x[/tex] ∈ {[tex]\displaystyle{ \frac{3\pi}{4} + k\pi, k \in Z} }[/tex]}
Explicație pas cu pas:
[tex]\displaystyle{ \frac{1}{1 + tg x} + \frac{1}{1 + ctg x} = 1 }[/tex]
[tex]\displaystyle{\frac{1}{1 + \frac{sinx}{cosx} } + \frac{1}{1+\frac{cosx}{sinx}} =1 }[/tex]
[tex]\displaystyle{ \frac{1}{\frac{cosx+sinx}{cosx}} + \frac{1}{\frac{sinx+cosx}{sinx}} = 1 }[/tex]
[tex]\displaystyle{ \frac{cosx}{cosx + sinx} + \frac{sinx}{cosx+sinx} = 1 }[/tex]
[tex]\displaystyle{ \frac{cosx+sinx}{cosx+sinx+cosx+sinx} = 1 }[/tex]
[tex]\displaystyle{ \frac{cosx+sinx}{2cosx+2sinx} = 1 }[/tex]
2cosx + 2sinx = cosx + sinx
cosx + sinx = 0
cosx = -sinx
[tex]\displaystyle{ x \in (0, \frac{\pi}{2} )}[/tex]
⇒ [tex]x[/tex] ∈ {[tex]\displaystyle{ \frac{3\pi}{4} + k\pi, k \in Z} }[/tex]}
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