How do you verify the identity csc^4theta-cot^4theta=2csc^2theta-1?

1 Answer
Apr 19, 2017

see below

Explanation:

Use the Pythagorean Identity
color(red)(cot^2 theta+1=csc^2 theta

color(red)(cot^2 theta=csc^2 theta-1--->isolate cot^2 theta

color(red)(1=csc^2 theta-cot^2 theta---> isolate 1

Left Hand Side:

color(red)(csc^4 theta-cot^4 theta=color(blue)((csc^2 theta+cot^2 theta)(csc^2 theta-cot^2 theta)

color(blue)(=(csc^2 theta+cot^2 theta)*1

color(blue)(=csc^2 theta+cot^2 theta

color(blue)(=csc^2 theta+csc^2 theta-1

color(blue)(=2csc^2 theta-1

color(blue)(=Right Hand Side