Rysunek pomocniczy:

Z funkcji trygonometrycznych otrzymujemy:   

$\frac{H}{r}=\mathrm{tg}\alpha$  

$H=r\mathrm{tg}\alpha$  

oraz

$\frac{H}{h}=\cos \beta$  

$\frac{H}{\cos \beta }=h$  

$\frac{r\mathrm{tg}\alpha }{\cos \beta }=h$  

oraz

$\frac{r}{l}=\cos \alpha$  

$\frac{r}{\cos \alpha }=l$  

 

Z tw. Pitagorasa otrzymujemy:

$h^2+x^2=l^2$  

${\left(\frac{r\mathrm{tg}\alpha }{\cos \beta }\right)}^2+x^2={\left(\frac{r}{\cos \alpha }\right)}^2$  

$\frac{r^2{\mathrm{tg}}^2\alpha }{{\cos }^2\beta }+x^2=\frac{r^2}{{\cos }^2\alpha }$  

$x^2=\frac{r^2}{{\cos }^2\alpha }-r^2\frac{{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }$  

$x^2=r^2\left(\frac{1}{{\cos }^2\alpha }-\frac{{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }\right)$  

$x^2=r^2\left(\frac{{\cos }^2\beta }{{\cos }^2\alpha {\cos }^2\beta }-\frac{{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }\right)$  

$x=r\sqrt{\frac{{\cos }^2\beta -{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }}$  

 

Zatem otrzymujemy:

$P=\frac{1}{2}\cdot 2x\cdot h$  

$P=x\cdot h$  

$P=r\sqrt{\frac{{\cos }^2\beta -{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }}\cdot \frac{r\mathrm{tg}\alpha }{\cos \beta }$  

$P=\frac{r^2\mathrm{tg}\alpha }{\cos \beta }\cdot \sqrt{\frac{{\cos }^2\beta -{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }}$  

 

Zauważmy, że:

$\frac{{\cos }^2\beta -{\sin }^2\alpha }{{\cos }^2\alpha {\cos }^2\beta }=\frac{{\cos }^2\beta -{\sin }^2\alpha \left({\cos }^2\beta +{\sin }^2\beta \right)}{{\cos }^2\alpha {\cos }^2\beta }=$  

$=\frac{{\cos }^2\beta -{\sin }^2\alpha {\cos }^2\beta -{\sin }^2\alpha {\sin }^2\beta }{{\cos }^2\alpha {\cos }^2\beta }=$  

$=\frac{\left(1-{\sin }^2\alpha \right){\cos }^2\beta -{\sin }^2\alpha {\sin }^2\beta }{{\cos }^2\alpha {\cos }^2\beta }=$  

$=\frac{{\cos }^2\alpha {\cos }^2\beta -{\sin }^2\alpha {\sin }^2\beta }{{\cos }^2\alpha {\cos }^2\beta }=$  

$=1-\frac{{\sin }^2\alpha {\sin }^2\beta }{cs{o}^2\alpha {\cos }^2\beta }=1-{\mathrm{tg}}^2\alpha {\mathrm{tg}}^2\beta$  

 

Zatem możemy zapisać:

$P=\frac{r^2\mathrm{tg}\alpha }{\cos \beta }\cdot \sqrt{1-{\mathrm{tg}}^2\alpha {\mathrm{tg}}^2\beta }$