CPM Homework Help : PC3 Problem 10-83

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10-83.

Verify the following trigonometric identities.

$\sec(x)-\cos(x)=$ $\tan(x)\sin(x)$
Step 1(a):
$\frac{1}{\cos(x)}-\cos(x)=$
Step 2(a):
$\frac{1}{\cos(x)}-\frac{\cos^2(x)}{\cos(x)}=$

$\sec(x)+\tan(x)=$ $\frac{1}{\sec(x)-\tan(x)}$
Hint (b):
In this case it is easier to work on the right side of the equation.
Step 1(b):
$=\frac{1}{\frac{1}{\cos(x)}-\frac{\sin(x)}{\cos(x)}}$
Step 2(b):
$=\frac{1}{\frac{1-\sin(x)}{\cos(x)}}$
Step 3(b):
$=\frac{\cos(x)}{1-\sin(x)}$
Step 4(b):
$=\frac{\cos(x)}{1-\sin(x)}\cdot\frac{1+\sin(x)}{1+\sin(x)}$

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