Zjednodušenie trigonometrického výrazu
Zjednoduš výraz \( \frac{1+\operatorname{cotg}^{2} x}{1+\operatorname{tg}^{2} x} \).
\( =\left(1+\operatorname{cotg}^{2} x\right): \frac{\operatorname{tg}^{2} x+1}{1}=\left(1+\operatorname{cotg}^{2} x\right) \cdot \frac{1}{\operatorname{tg}^{2} x+1}=\operatorname{tg}^{2} x \)
\( =\left(1+\operatorname{cotg}^{2} x\right): \frac{\operatorname{tg}^{2} x+1}{\operatorname{tg}^{2} x}=\left(1+\operatorname{cotg}^{2} x\right) \cdot \frac{\operatorname{tg}^{2} x}{\operatorname{tg}^{2} x+1}=\operatorname{tg}^{2} x \)
\( =\left(1+\operatorname{cotg}^{2} x\right): \frac{\operatorname{cotg}^{2} x+1}{\operatorname{cotg}^{2} x}=\left(1+\operatorname{cotg}^{2} x\right) \cdot \frac{\operatorname{cotg}^{2} x}{\operatorname{cotg}^{2} x+1}=\operatorname{cotg}^{2} x \)
\( =\left(1+\operatorname{cotg}^{2} x\right): \frac{1+\operatorname{cotg}^{2} x}{\operatorname{tg}^{2} x}=\left(1+\operatorname{cotg}^{2} x\right) \cdot \frac{1}{\operatorname{tg}^{2} x}=1 \)