How do you verify (1+csc(x))/(cot(x)+cos(x)) = sec(x)?

May 21, 2016

On the left side,

$\frac{1 + \csc x}{\cot x + \cos x}$

Rewrite $\csc x$ and $\cot x$ in terms of $\sin x$ and $\cos x$.

$= \frac{1 + \textcolor{b l u e}{\frac{1}{\sin x}}}{\textcolor{p u r p \le}{\frac{\cos x}{\sin x}} + \cos x}$

Simplify.

$= \frac{\frac{\sin x + 1}{\sin x}}{\frac{\cos x + \sin x \cos x}{\sin x}}$

$= \frac{\sin x + 1}{\sin x} \cdot \frac{\sin x}{\cos x + \sin x \cos x}$

$= \frac{\sin x + 1}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\sin x}}}} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\sin x}}}}{\cos x + \sin x \cos x}$

$= \frac{\sin x + 1}{\cos x + \sin x \cos x}$

Factor out $\cos x$ from the denominator.

$= \frac{\sin x + 1}{\cos x \left(1 + \sin x\right)}$

Simplify.

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\sin x + 1}}}}{\cos x \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(1 + \sin x\right)}}}}$

$= \frac{1}{\cos} x$

$= \sec x$