How do you prove #csc^2(theta)(1-cos^2(theta)=1#?

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Answer 1 · 2016-05-04T19:18:25

see below

Left Side:#=csc^2 theta (1-cos^2 theta)#

#= csc^2 theta * sin^2 theta#

#= 1/sin^2 theta * sin^2 theta#

#=1#

#=# Right Side

Answer 2 · 2016-05-04T19:22:25

First, we start with

#csc^2theta(1-cos^2theta) = 1,#

and consider what identities make sense to use. Note that #csctheta = 1/sintheta#. Thus, #csc^2theta = 1/sin^2theta#.

That means it would be convenient to have #csc^2theta*sin^2theta#, since it would cancel to give #1#.

#sin^2theta + cos^2theta = 1#

is a trig identity we can use. Therefore, we have

#csc^2theta(sin^2theta) = 1/cancel(sin^2theta)cancel(sin^2theta) = color(blue)(1)#