Solution
Step 1
Trigonometric inverse theorem
[tex]csc\theta\text{ = }\frac{1}{sin\theta}\text{ TRUE}[/tex]
Step 2
[tex]\begin{gathered} Trigonometric\text{ ratio} \\ cot\theta\text{ = }\frac{cos\theta}{sin\theta} \\ Option\text{ B is FALSE} \end{gathered}[/tex]
Step 3
[tex]\begin{gathered} 1\text{ + cot}^2\theta \\ =\text{ 1 + }\frac{cos^2\theta}{sin^2\theta} \\ =\text{ }\frac{sin^2\theta+cos^2\theta}{sin^2\theta} \\ \text{= }\frac{1}{sin^2\theta} \\ \text{= csc}^2\theta \\ 1\text{ + cot}^2\theta\text{ = csc}^2\theta\text{ TRUE} \end{gathered}[/tex]
Step 4
[tex]\begin{gathered} sin(-\pi) \\ =\text{ sin\lparen0-}\pi) \\ =\text{ sin0cos}\pi\text{ - sin}\pi cos0 \\ =\text{ -sin}\pi \\ sin(-\pi)\text{ = -sin}\pi \\ Option\text{ D is FALSE} \end{gathered}[/tex]
Step 5
[tex]\begin{gathered} cos(-\pi) \\ =\text{ cos\lparen0-}\pi) \\ =\text{ cos0cos}\pi\text{ + sin0sin}\pi \\ =\text{ cos}\pi \\ cos(-\pi)\text{ = cos}\pi\text{ TRUE} \end{gathered}[/tex]
