Question

Simplify this into 1 trigonometry function:

(sec(x)-cos(x))/sin(x)


Hint: The answer is tan(x) but how do you get this?

Answer 1

`\bf sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta) \\\\\\ tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)}\qquad \qquad sec(\theta)=\cfrac{1}{cos(\theta)}\\\\ -------------------------------\\\\ \cfrac{sec(x)-cos(x)}{sin(x)}\implies \cfrac{(1)/(cos(x))-cos(x)}{sin(x)}\implies \cfrac{(1-cos^2(x))/(cos(x))}{sin(x)}\implies \cfrac{(sin^2(x))/(cos(x))}{(sin(x))/(1)} \\\\\\`


`\bf \cfrac{\underline{sin^2(x)}}{cos(x)}\cdot \cfrac{1}{\underline{sin(x)}}\implies \cfrac{sin(x)}{cos(x)}\implies tan(x)`