CPM Homework Help : PCT Problem 4-98

Simplify the following trig expressions.

$\frac { \operatorname { cos } x } { \operatorname { sec } x } + \frac { \operatorname { sin } x } { \operatorname { csc } x }$

$\frac { \operatorname { tan } x + \operatorname { cot } x } { \operatorname { sec } x \operatorname { csc } x }$

Steps (a):

1. Change the $\text{sec }x$ and $\text{csc }x$ to forms of the sine and cosine.
2. Simplify.
3. Substitute using the Fundamental Pythagorean Identities.

Step 1(b):

Step 2(b):

Step 3(b):

$\frac{\frac{\text{sin}x)}{\text{cos}(x)}+\frac{\text{cos}(x)}{\text{sin}(x)}}{\text{sec}(x)\text{csc}(x)}$

$\small \frac{\frac{\text{sin}^2(x)+\text{cos}^2(x)}{\text{cos}(x)\text{sin}(x)}}{\frac{1}{\text{cos}(x)sin(x)}}=1$

$\small\frac{\frac{1}{\text{cos}(x)\text{sin}(x)}}{\frac{1}{\text{cos}(x)\text{sin}(x)}}=1$