SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given values
[tex]\begin{gathered} \sin 30^{\circ}=\frac{1}{2} \\ \tan 30^{\circ}=\frac{\sqrt[]{3}}{3} \end{gathered}[/tex]STEP 2: Get the value of cosec 30 degrees
[tex]\begin{gathered} \text{csc }\theta\text{ means cosec }\theta\text{ and it is the inverse of sin }\theta \\ \csc \theta=\frac{1}{\sin \theta} \\ \sin 30=\frac{1}{2} \\ \csc 30=\frac{1}{\sin 30} \\ By\text{ substituting 1/2 for sin 30} \\ \csc 30=\frac{1}{\frac{1}{2}}=1\times2=2 \end{gathered}[/tex]Therefore, the value of csc 30 is 2
STEP 3: Get the value of cot 60
[tex]\begin{gathered} \cot \theta=\frac{1}{\tan \theta} \\ \tan 60=\sqrt[]{3} \\ \cot 60=\frac{1}{\sqrt[]{3}} \\ By\text{ rationalization,} \\ \frac{1}{\sqrt[]{3}}\times\frac{\sqrt[]{3}}{\sqrt[]{3}}=\frac{\sqrt[]{3}}{3} \end{gathered}[/tex]Hence, the value of cot 60 is √3/3
STEP 4: Get the value of cos 30
[tex]\begin{gathered} \frac{\sin\theta}{\tan\theta}=\cos \theta \\ \cos 30=\frac{\sin 30}{\tan 30} \\ By\text{ substitution,} \\ \cos 30=\frac{1}{2}\div\frac{\sqrt[]{3}}{3} \\ \Rightarrow\frac{1}{2}\times\frac{3}{\sqrt[]{3}}=\frac{3}{2\sqrt[]{3}} \\ By\text{ rationalization,} \\ \cos 30=\frac{3}{2\sqrt[]{3}}\times\frac{2\sqrt[]{3}}{2\sqrt[]{3}}=\frac{3\times2\sqrt[]{3}}{2\times2\times3}=\frac{6\sqrt[]{3}}{12}=\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]Hence, the value of cos 30 is √3/2
STEP 5: Get the value of cot 30
[tex]\begin{gathered} \cot \theta=\frac{1}{\tan \theta} \\ \cot 30=\frac{1}{\tan 30} \\ By\text{ substitution,} \\ \cot 30=\frac{1}{\frac{\sqrt[]{3}}{3}} \\ \cot 30=1\div\frac{\sqrt[]{3}}{3}=1\times\frac{3}{\sqrt[]{3}}=\frac{3}{\sqrt[]{3}} \\ By\text{ rationalization,} \\ \frac{3}{\sqrt[]{3}}=\frac{3}{\sqrt[]{3}}\times\frac{\sqrt[]{3}}{\sqrt[]{3}}=\frac{3\sqrt[]{3}}{3}=\sqrt[]{3} \end{gathered}[/tex]Hence, the value of cot 30 is √3
