Question

Simplify the expression into a single trig function.

Step by step explanation please!

Answer 1

`sin^2(\theta)+cos^2(\theta)=1\implies sin^2(\theta)=1-cos^2(\theta) \\\\\\ sec(\theta)=\cfrac{1}{cos(\theta)}\hspace{5em}cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)} \\\\[-0.35em] ~\dotfill`

`\cfrac{sin(t)}{sec(t)-cos(t)}\implies \cfrac{sin(t)}{(1)/(cos(t))-cos(t)}\implies \cfrac{sin(t)}{(1-cos^2(t))/(cos(t))} \\\\\\ \cfrac{sin(t)}{1}\cdot \cfrac{cos(t)}{1-cos^2(t)}\implies \cfrac{\stackrel{1}{~~\begin{matrix} sin(t) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ }}{1}\cdot \cfrac{cos(t)}{\underset{sin(t)}{~~\begin{matrix} sin^2(t) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ }}\implies \cfrac{cos(t)}{sin(t)}\implies cot(t)`