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

May 4, 2016

see below

#### Explanation:

Left Side:$= {\csc}^{2} \theta \left(1 - {\cos}^{2} \theta\right)$

$= {\csc}^{2} \theta \cdot {\sin}^{2} \theta$

$= \frac{1}{\sin} ^ 2 \theta \cdot {\sin}^{2} \theta$

$= 1$

$=$ Right Side

May 4, 2016

${\csc}^{2} \theta \left(1 - {\cos}^{2} \theta\right) = 1 ,$

and consider what identities make sense to use. Note that $\csc \theta = \frac{1}{\sin} \theta$. Thus, ${\csc}^{2} \theta = \frac{1}{\sin} ^ 2 \theta$.

That means it would be convenient to have ${\csc}^{2} \theta \cdot {\sin}^{2} \theta$, since it would cancel to give $1$.

${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

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

${\csc}^{2} \theta \left({\sin}^{2} \theta\right) = \frac{1}{\cancel{{\sin}^{2} \theta}} \cancel{{\sin}^{2} \theta} = \textcolor{b l u e}{1}$