Answer
$csc\phi$ $+cot\phi$ $=\frac{sin\phi}{1-cos \phi}$
Work Step by Step
We need to prove the identity
$csc\phi$ $+cot\phi$ $=\frac{sin\phi}{1-cos \phi}$
Let us solve the left side of the given identity.
$csc\phi$ $+cot\phi$ $=\frac{1}{sin\phi}+\frac{cos \phi}{sin\phi}$
$=\frac{1+cos \phi}{sin\phi}$
$=\frac{1+cos \phi}{sin\phi}\times \frac{1-cos \phi}{1-cos\phi}$
$=\frac{1-cos^{2} \phi}{sin\phi(1-cos\phi)}$
$=\frac{sin^{2} \phi}{sin\phi(1-cos\phi)}$
Hence, $csc\phi$ $+cot\phi$ $=\frac{sin\phi}{1-cos \phi}$
