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

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

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