Zadanie 1
[tex]\text{L}=(\text{ctg}^2\alpha+1)\cdot\sin^2\alpha=\Big[\Big(\dfrac{\cos\alpha}{\sin\alpha}\Big)^2+1\Big]\cdot\sin^2\alpha=\\\\=\Big(\dfrac{\cos^2\alpha}{\sin^2\alpha}+\dfrac{\sin^2\alpha}{\sin^2\alpha}\Big)\cdot\sin^2\alpha=\dfrac{\cos^2\alpha+\sin^2\alpha}{\sin^2\alpha}\cdot\sin^2\alpha=\\\\\\=\cos^2\alpha+\sin^2\alpha=1=\text{P}[/tex]
Zadanie 2
[tex]\text{P}=\sin\alpha\cdot\text{tg }\alpha=\sin\alpha\cdot\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{\sin^2\alpha}{\cos\alpha}=\\\\=\dfrac{\sin^2\alpha+\cos^2\alpha-\cos^2\alpha}{\cos\alpha}=\dfrac{1-\cos^2\alpha}{\cos\alpha}=\\\\=\dfrac{1}{\cos\alpha}-\cos\alpha=\text{L}[/tex]
Zadanie 3
[tex]\text{L}=\dfrac{\text{tg }\alpha}{\text{tg }\alpha+\text{ctg }\alpha}=\dfrac{\frac{\sin\alpha}{\cos\alpha}}{\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha}}=\dfrac{\frac{\sin\alpha}{\cos\alpha}}{\frac{\sin^2\alpha}{\sin\alpha\cos\alpha}+\frac{\cos^2\alpha}{\sin\alpha\cos\alpha}}=[/tex]
[tex]\dfrac{\frac{\sin\alpha}{\cos\alpha}}{\frac{\sin^2\alpha+\cos^2\alpha}{\sin\alpha\cos\alpha}}=\dfrac{\frac{\sin\alpha}{\cos\alpha}}{\frac{1}{\sin\alpha\cos\alpha}}=\dfrac{\sin^2\alpha\cos\alpha}{\cos\alpha}=\sin^2\alpha=\text{P}[/tex]
