Question

Simplify the expression 1/√(1-cos^2 x) . Remember cos x= 1/csc x

Answer 1

First, we are going to use the Pythagorean identity: $sin^2(x)+cos^2(x)=1$ to rewrite our expression. Solving for $sin^2(x)$ in our Pythagorean identity  we get that $sin^2(x)=1-cos^2(x)$, so we can rewrite our expression as follows:
$\frac{1}{ \sqrt{1-cos^2(x)} } = \frac{1}{ \sqrt{sin^2(x)} } = \frac{1}{sin(x)}$

Next, we are going to use the trig identity: $csc(x)= \frac{1}{sin(x)}$ to completely simplify our expression:
$\frac{1}{sin(x)} =csc(x)$

We can conclude that $\frac{1}{ \sqrt{1-cos^2(x)} }=csc(x)$