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

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We wish to show that
$$
\frac{\cos y}{1+\sin y}+\frac{1+\sin y}{\cos y}=\frac{2}{\cos y}.
$$

Step 1. Write both terms with a common denominator $$\cos y(1+\sin y)$$:
$$
\frac{\cos y}{1+\sin y}+\frac{1+\sin y}{\cos y}
=\frac{\cos^2y+(1+\sin y)^2}{\cos y(1+\sin y)}.
$$

Step 2. Expand the numerator. Notice that
$$
(1+\sin y)^2=1+2\sin y+\sin^2 y.
$$
Thus the numerator becomes:
$$
\cos^2y+1+2\sin y+\sin^2 y.
$$

Step 3. Use the Pythagorean identity $$\cos^2 y+\sin^2 y=1$$ to combine like terms:
$$
\cos^2y+\sin^2y=1,
$$
so the numerator is:
$$
1+1+2\sin y = 2(1+\sin y).
$$

Step 4. Substitute back into the fraction:
$$
\frac{2(1+\sin y)}{\cos y(1+\sin y)}.
$$

Step 5. Cancel the common factor $$(1+\sin y)$$ (provided $$1+\sin y\neq0$$):
$$
\frac{2(1+\sin y)}{\cos y(1+\sin y)}=\frac{2}{\cos y}.
$$

Thus, we have shown that
$$
\frac{\cos y}{1+\sin y}+\frac{1+\sin y}{\cos y}=\frac{2}{\cos y}.
$$
$$ \frac{\cos y}{1+\sin y}+\frac{1+\sin y}{\cos y}=\frac{2}{\cos y} $$

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