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To prove that 

\[ \frac{\cot y (1 + \tan^2 y)}{\csc y} = \sec y \]

we start by using trigonometric identities.

1. Recall the identity for \(\cot y\):
   \[ \cot y = \frac{1}{\tan y} \]

2. Recall the Pythagorean identity:
   \[ 1 + \tan^2 y = \sec^2 y \]

3. Recall the identity for \(\csc y\):
   \[ \csc y = \frac{1}{\sin y} \]

4. Recall the identity for \(\sec y\):
   \[ \sec y = \frac{1}{\cos y} \]

Now, substitute these identities into the left-hand side of the given equation:

\[ \frac{\cot y (1 + \tan^2 y)}{\csc y} \]

Replace \(\cot y\) and \(1 + \tan^2 y\):

\[ \frac{\frac{1}{\tan y} \cdot \sec^2 y}{\csc y} \]

Next, replace \(\tan y\) with \(\frac{\sin y}{\cos y}\) and \(\sec^2 y\) with \(\frac{1}{\cos^2 y}\):

\[ \frac{\frac{1}{\frac{\sin y}{\cos y}} \cdot \frac{1}{\cos^2 y}}{\csc y} = \frac{\frac{\cos y}{\sin y} \cdot \frac{1}{\cos^2 y}}{\csc y} \]

Simplify the fractions:

\[ \frac{\frac{\cos y}{\sin y \cos^2 y}}{\csc y} = \frac{\frac{1}{\sin y \cos y}}{\csc y} \]

Replace \(\csc y\) with \(\frac{1}{\sin y}\):

\[ \frac{\frac{1}{\sin y \cos y}}{\frac{1}{\sin y}} \]

Simplify the division:

\[ \frac{1}{\sin y \cos y} \cdot \sin y = \frac{\sin y}{\sin y \cos y} = \frac{1}{\cos y} = \sec y \]

Thus, we have shown that

\[ \frac{\cot y (1 + \tan^2 y)}{\csc y} = \sec y \]

This completes the proof.

Do you want more details or have any questions? 

Here are some other questions you might find interesting:
1. How do you prove other trigonometric identities?
2. Can you solve an equation involving \(\sin\) and \(\cos\)?
3. What are some applications of trigonometric identities?
4. How do you convert between different trigonometric functions?
5. How can you use trigonometric identities to simplify expressions?

**Tip:** When proving trigonometric identities, always start by writing down the fundamental identities you know and see how they can be applied to the given equation.

Learn how to prove the trigonometric identity involving cotangent, secant, cosecant, and tangent. This problem demonstrates the use of trigonometric identities such as the Pythagorean identity and definitions of trigonometric functions. Suitable for high school students studying trigonometry.

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