Question

PLS HELP, FACE PARTE DIN NOILE TESTE DE ANTRENAMENT PENTRU BAC, PROFIL MATE-INFO

Answer 1

Răspuns:

$x$ ∈ {$\displaystyle{ \frac{3\pi}{4} + k\pi, k \in Z} }$}

Explicație pas cu pas:

$\displaystyle{ \frac{1}{1 + tg x} + \frac{1}{1 + ctg x} = 1 }$

$\displaystyle{\frac{1}{1 + \frac{sinx}{cosx} } + \frac{1}{1+\frac{cosx}{sinx}} =1 }$

$\displaystyle{ \frac{1}{\frac{cosx+sinx}{cosx}} + \frac{1}{\frac{sinx+cosx}{sinx}} = 1 }$

$\displaystyle{ \frac{cosx}{cosx + sinx} + \frac{sinx}{cosx+sinx} = 1 }$

$\displaystyle{ \frac{cosx+sinx}{cosx+sinx+cosx+sinx} = 1 }$

$\displaystyle{ \frac{cosx+sinx}{2cosx+2sinx} = 1 }$

2cosx + 2sinx = cosx + sinx

cosx + sinx = 0

cosx = -sinx

$\displaystyle{ x \in (0, \frac{\pi}{2} )}$

⇒ $x$ ∈ {$\displaystyle{ \frac{3\pi}{4} + k\pi, k \in Z} }$}

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