Question

Verify identity:

(sec(x)-csc(x))/(sec(x)+csc(x))=(tan(x)-1)/(tan(x)+1)

Answer 1

so hmmm let's do the left-hand-side first


`\bf \cfrac{sec(x)-csc(x)}{sec(x)+csc(x)}\implies \cfrac{(1)/(cos(x))-(1)/(sin(x))}{(1)/(cos(x))+(1)/(sin(x))}\implies \cfrac{(sin(x)-cos(x))/(cos(x)sin(x))}{(sin(x)+cos(x))/(cos(x)sin(x))} \\\\\\ \cfrac{sin(x)-cos(x)}{cos(x)sin(x)}\cdot \cfrac{cos(x)sin(x)}{sin(x)+cos(x)}\implies \boxed{\cfrac{sin(x)-cos(x)}{sin(x)+cos(x)}}`

now, let's do the right-hand-side then


`\bf \cfrac{tan(x)-1}{tan(x)+1}\implies \cfrac{(sin(x))/(cos(x))-1}{(sin(x))/(cos(x))+1}\implies \cfrac{(sin(x)-cos(x))/(cos(x))}{(sin(x)+cos(x))/(cos(x))} \\\\\\ \cfrac{sin(x)-cos(x)}{cos(x)}\cdot \cfrac{cos(x)}{sin(x)+cos(x)}\implies \boxed{\cfrac{sin(x)-cos(x)}{sin(x)+cos(x)}}`