How do you simplify the following trigonometric expression: (( sec(x)*sin(x)+cos( \pi-x) ) /(1+sce(x)) Thanks!!

Question

How do you simplify the following trigonometric expression:

(
`( sec(x)*sin(x)+cos( \pi-x) ) /(1+sce(x))`

Thanks!!

Answer 1

`\bf \cfrac{sec(x)sin(x)+cos(\pi -x)}{1+sec(x)}\\\\ -----------------------------\\\\ sec(\theta)=\cfrac{1}{cos(\theta)}\qquad \qquad cos(\pi )=-1\qquad sin(\pi )=0 \\\\ cos({{ \alpha}} + {{ \beta}})= cos({{ \alpha}})cos({{ \beta}})- sin({{ \alpha}})sin({{ \beta}})\\\\ -----------------------------\\\\ \cfrac{(1)/(cos(x))sin(x)+[cos(\pi )cos(x)+sin(\pi )sin(x)]}{1+(1)/(cos(x))} \\\\\\ \cfrac{(sin(x))/(cos(x))+[-1\cdot cos(x)+0\cdot sin(x)]}{(cos(x)+1)/(cos(x))}`


`\bf \cfrac{(sin(x))/(cos(x))+[-cos(x)]}{(cos(x)+1)/(cos(x))}\implies \cfrac{(sin(x)-cos^2(x))/(cos(x))}{(cos(x)+1)/(cos(x))} \\\\\\ \cfrac{sin(x)-cos^2(x)}{cos(x)}\cdot \cfrac{cos(x)}{cos(x)+1}\implies \cfrac{sin(x)-cos^2(x)}{cos(x)+1}`