Verify Trig Identity

Question

(sec(x)+1)/(sec(x)-1)=cot²(x/2)

Answer 1

Prove (sec(x)+1)/(sec(x)-1) = cot²(x/2)

substitute sec x = 1/ cos x

( 1/cos(x) + 1) / (1 /cos(x) -1) , add the fractions in each side and you get

( ( 1+ cos(x) / cos(x)) / ( ( 1- cos(x) / cos(x)) = [(1+ cos(x) / cos(x))] .[ cos(x) / 1- cos ( x))] = 1+cos(x) / 1-cos(x)

From trigonometric identities tan( x/2) = ±√1+cos(x) / 1-cos(x) , square both sides

tan² ( x/2) = 1+cos(x) / 1-cos(x) , if cot(x) = 1/ tan(x), then cot² ( x/2) = 1/ tan²(x/2)

and cot² ( x/2) = 1 / [ 1+cos(x) / 1-cos(x)] = 1- cos(x) / 1+ cos(x) = (sec(x)+1)/(sec(x)-1)