Answer :
To verify the identity \(\frac{\cot ^2 t}{\csc t}=\cot t \cos t\), let's break down the steps clearly and determine which statement correctly establishes the identity.
1. Express cotangent and cosecant in terms of sine and cosine:
[tex]\[ \cot t = \frac{\cos t}{\sin t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
2. Square the cotangent function:
[tex]\[ \cot^2 t = \left(\frac{\cos t}{\sin t}\right)^2 = \frac{\cos^2 t}{\sin^2 t} \][/tex]
3. Substitute thus-derived expressions into the original identity:
[tex]\[ \frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \times \sin t = \frac{\cos^2 t}{\sin t} \][/tex]
5. Combine the expressions:
[tex]\[ \frac{\cos^2 t}{\sin t} = \left(\frac{\cos t}{\sin t}\right) \cos t = \cot t \cos t \][/tex]
Therefore, [tex]\[ \frac{\cot ^2 t}{\csc t} = \cot t \cos t \][/tex]
Among the given options, the one that correctly establishes this identity is:
A. \(\frac{\cot ^2 t}{\csc t}=\cot t\left(\frac{\cos t}{\sin t}\right)(\sin t)=\cot t \cos t\)
This statement methodically breaks down the required steps, showing that dividing [tex]\(\cot^2 t\)[/tex] by [tex]\(\csc t\)[/tex] and then simplifying gives [tex]\(\cot t \cos t\)[/tex], thus establishing the identity.
1. Express cotangent and cosecant in terms of sine and cosine:
[tex]\[ \cot t = \frac{\cos t}{\sin t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
2. Square the cotangent function:
[tex]\[ \cot^2 t = \left(\frac{\cos t}{\sin t}\right)^2 = \frac{\cos^2 t}{\sin^2 t} \][/tex]
3. Substitute thus-derived expressions into the original identity:
[tex]\[ \frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \times \sin t = \frac{\cos^2 t}{\sin t} \][/tex]
5. Combine the expressions:
[tex]\[ \frac{\cos^2 t}{\sin t} = \left(\frac{\cos t}{\sin t}\right) \cos t = \cot t \cos t \][/tex]
Therefore, [tex]\[ \frac{\cot ^2 t}{\csc t} = \cot t \cos t \][/tex]
Among the given options, the one that correctly establishes this identity is:
A. \(\frac{\cot ^2 t}{\csc t}=\cot t\left(\frac{\cos t}{\sin t}\right)(\sin t)=\cot t \cos t\)
This statement methodically breaks down the required steps, showing that dividing [tex]\(\cot^2 t\)[/tex] by [tex]\(\csc t\)[/tex] and then simplifying gives [tex]\(\cot t \cos t\)[/tex], thus establishing the identity.
