Prove that: $$ \frac{\cos y}{1+\sin y}=\frac{1+\sin y}{\cos y}=\frac{2}{\cos y} $$

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Verify the identity by following steps:
- step0: Verify:
  $$\frac{1+\sin\left(y\right)}{\cos\left(y\right)}=\frac{2}{\cos\left(y\right)}$$
- step1: Choose a side to work on:
  $$\frac{1}{\cos\left(y\right)}+\frac{\sin\left(y\right)}{\cos\left(y\right)}=\frac{2}{\cos\left(y\right)}$$
- step2: Verify the identity:
  $$\textrm{false}$$
Determine whether the expression $$ \frac{\cos y}{1+\sin y}=\frac{1+\sin y}{\cos y} $$ is always true.
Verify the identity by following steps:
- step0: Verify:
  $$\frac{\cos\left(y\right)}{1+\sin\left(y\right)}=\frac{1+\sin\left(y\right)}{\cos\left(y\right)}$$
- step1: Choose the other side to work on:
  $$\frac{\cos\left(y\right)}{1+\sin\left(y\right)}=\frac{1}{\cos\left(y\right)}+\frac{\sin\left(y\right)}{\cos\left(y\right)}$$
- step2: Verify the identity:
  $$\textrm{false}$$
The statement $$ \frac{\cos y}{1+\sin y}=\frac{1+\sin y}{\cos y}=\frac{2}{\cos y} $$ is not true.

1. **First part:** $$ \frac{\cos y}{1+\sin y} = \frac{1+\sin y}{\cos y} $$

This equality is false. To see why, we can cross-multiply:
   $$
   \cos^2 y = (1 + \sin y)(1 + \sin y) = 1 + 2\sin y + \sin^2 y
   $$
   This simplifies to:
   $$
   \cos^2 y = 1 + 2\sin y + \sin^2 y
   $$
   Using the Pythagorean identity $$ \cos^2 y + \sin^2 y = 1 $$, we can substitute $$ \cos^2 y $$ with $$ 1 - \sin^2 y $$:
   $$
   1 - \sin^2 y = 1 + 2\sin y + \sin^2 y
   $$
   Rearranging gives:
   $$
   -\sin^2 y - 2\sin y = 0
   $$
   Factoring out $$-\sin y$$:
   $$
   -\sin y(\sin y + 2) = 0
   $$
   This implies $$ \sin y = 0 $$ or $$ \sin y = -2 $$ (which is not possible). Thus, the equality does not hold for all $$ y $$.

2. **Second part:** $$ \frac{1+\sin y}{\cos y} = \frac{2}{\cos y} $$

This equality is also false. Cross-multiplying gives:
   $$
   1 + \sin y = 2
   $$
   This simplifies to:
   $$
   \sin y = 1
   $$
   This is only true for specific values of $$ y $$ (e.g., $$ y = \frac{\pi}{2} + 2k\pi $$ for integers $$ k $$), not for all $$ y $$.

In conclusion, the original statement is not universally true.
The given equation $$ \frac{\cos y}{1+\sin y}=\frac{1+\sin y}{\cos y}=\frac{2}{\cos y} $$ is not true for all values of $$ y $$.

Prove that:

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