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Question Number 188270 by mnjuly1970 last updated on 27/Feb/23

Commented by mnjuly1970 last updated on 27/Feb/23

$$inA\stackrel{\mathrm{\Delta }}{BC}prove:$$$$\frac{1}{cos\left(A\right)}+\frac{1}{cos\left(B\right)}+\frac{1}{cos\left(C\right)}⩾6$$$$$$$$note:0<A,B,C<{90}^{°}$$$$$$

Answered by mahdipoor last updated on 27/Feb/23

$$f(x,y,z)=\frac{1}{cos\left(x\right)}+\frac{1}{cos\left(y\right)}+\frac{1}{cos\left(z\right)}$$$$g(x,y,z)=x+y+z=180\left(eqI\right)$$$$p=(a,b,c),f\left(p\right)=minf$$$$\Rightarrow ▽\overrightarrow{f}\left(p\right)=\lambda ▽\overrightarrow{g}\left(p\right)$$$$\Rightarrow (-\frac{sin\left(a\right)}{{cos}^2\left(a\right)})i+(-\frac{sin\left(b\right)}{{cos}^2\left(b\right)})j+(-\frac{sin\left(c\right)}{{cos}^2\left(c\right)})k=$$$$=\lambda (i+j+k)\left(eqII\right)$$$$\Rightarrow \Rightarrow eqI\mathrm{\&}II\Rightarrow$$$$\frac{sina}{{cos}^{2}a}=\frac{sinb}{{cos}^{2}b}=\frac{sinc}{{cos}^{2}c}=-\lambda$$$$a+b+c=180$$$$\Rightarrow \Rightarrow a=b=c=60\Rightarrow minf=\frac{3}{cos\left(60\right)}=6$$

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