a)
[tex]\cos\alpha*\left(\frac{1}{\cos\alpha}-\cos\alpha\right)=\sin^2\alpha\\
\cos\alpha*\left(\frac{1}{\cos\alpha}-\cos\alpha\right)=1-\cos^2\alpha=sin^2\alpha[/tex]
Jest to tożsamość trygonometryczna.
b)
[tex]1-\sin\alpha=\frac{\text{ctg }\alpha-\cos\alpha}{\text{ctg }\alpha}\\
\frac{\text{ctg }\alpha-\cos\alpha}{\text{ctg }\alpha}=\frac{\text{ctg }\alpha}{\text{ctg }\alpha}-\frac{\cos\alpha}{\text{ctg }\alpha}=1-\frac{\cos\alpha}{\frac{\cos\alpha}{\sin\alpha}}=1-\cos\alpha*\frac{\sin\alpha}{\cos\alpha}=1-\sin\alpha[/tex]
c)
[tex]\frac{2}{\sin^2\alpha}-1=1+2*\text{ctg}^2\alpha\\
1+2*\text{ctg}^2\alpha=1+2*\frac{\cos^2\alpha}{\sin&^2\alpha}=1+2*\frac{1-\sin^2\alpha}{\sin&^2\alpha}=1+2*(\frac{1}{\sin&^2\alpha}-\frac{\sin&^2\alpha}{\sin&^2\alpha})=1+2*(\frac{1}{\sin&^2\alpha}-1)=1+\frac{2}{\sin&^2\alpha}-2=\frac{2}{\sin&^2\alpha}-1[/tex]
