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4-45.

Given $x^2 + y^2 = 1$ , rewrite each of the following expressions in terms of only $x$ .

$\frac { y ^ { 2 } } { x } + x$
Step (a):
$\frac{y^2}{x}+\frac{x^2}{x}$

$\frac { y ^ { 2 } } { \frac { 1 } { x ^ { 2 } } - 1 }$
Step (b):
$\frac{\frac{y^2}{1}}{\frac{1}{x^2}-\frac{1}{1}}\cdot\frac{\frac{x^2}{1}}{\frac{x^2}{1}}$
Hint (b):
$1-x^2=y^2$

$\frac { 1 } { 1 - y } + \frac { 1 } { 1 + y }$
Step (c):
$\frac{1+y}{(1-y)(1+y)}+\frac{1-y}{(1-y)(1+y)}$
Hint (c):
$1-y^2=x^2$

$\frac { 1 } { \frac { 1 } { x } - \frac { y } { x } } + \frac { 1 } { \frac { 1 } { x } + \frac { y } { x } }$
Steps (d):
$\small \frac{1}{\frac{1-y}{x}}+\frac{1}{\frac{1+y}{x}}$

$\frac{x}{1-y}+\frac{x}{1+y}$

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