Bonjour;
Exercice n° 18 .
[tex] cosec(\theta) + cot(\theta) = \dfrac{1}{sin(\theta)} + \dfrac{cos(\theta)}{sin(\theta)} = \dfrac{1+cos(\theta)}{sin(\theta)} \ ;\\\\\\ \Rightarrow (cosec(\theta) + cot(\theta))^2 = \dfrac{(1+cos(\theta))^2}{sin^2(\theta)} = \dfrac{(1+cos(\theta))^2}{1-cos^2(\theta)}\\\\\\ = \dfrac{(1+cos(\theta))^2}{(1-cos(\theta))(1+cos(\theta))} = \dfrac{1+cos(\theta)}{1-cos(\theta)} \ . [/tex]
Exercice n° 19 .
[tex] \dfrac{sin(\theta)}{1+sin(\theta)} + \dfrac{1-sin(\theta)}{sin(\theta)} = \dfrac{sin^2(\theta)+(1+sin(\theta))(1-sin(\theta))}{sin(\theta)(1+sin(\theta))}\\\\\\ = \dfrac{sin^2(\theta)+1-sin^2(\theta)}{sin(\theta)(1+sin(\theta))} = \dfrac{1}{sin(\theta)(1+sin(\theta))} = \dfrac{cosec(\theta)}{1+sin(\theta)} \ . [/tex]
