CPM Homework Help : PC3 Problem 9-63

Verify that $\frac { 1 } { 1 + \tan ^ { 2 } ( x ) } + \frac { 1 } { 1 + \cot ^ { 2 } ( x ) } = 1$ by simplifying the left side and using a Pythagorean identity.

Step 1:

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

Step 2:

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

Step 3:

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