First, we are going to use the Pythagorean identity: [tex]sin^2(x)+cos^2(x)=1[/tex] to rewrite our expression. Solving for [tex]sin^2(x)[/tex] in our Pythagorean identity we get that [tex]sin^2(x)=1-cos^2(x)[/tex], so we can rewrite our expression as follows:
[tex] \frac{1}{ \sqrt{1-cos^2(x)} } = \frac{1}{ \sqrt{sin^2(x)} } = \frac{1}{sin(x)} [/tex]
Next, we are going to use the trig identity: [tex]csc(x)= \frac{1}{sin(x)} [/tex] to completely simplify our expression:
[tex] \frac{1}{sin(x)} =csc(x)[/tex]
We can conclude that [tex] \frac{1}{ \sqrt{1-cos^2(x)} }=csc(x)[/tex]
