Question

Prove that (CotX/(cosecX+1))+((cosecX+1)/cotX)=2secX

Answer 1

Answer:

hope it help:)

Step-by-step explanation:

csc^2(x) = 1+cot^2(x)

(csc(x)+1)(csc(x)-1) = csc^2(x) - 1 = cot^2(x)

So cot(x)/(csc(x)+1)*(csc(x)-1)/(csc(x)-1) = cot(x)(csc(x)-1)/(csc^2(x)-1) = cot(x)(csc(x)-1)/(cot^2(x)) = (csc(x)-1)/cot(x)

so our left hand side becomes

(csc(x)-1)/cot(x) + (csc(x)+1)/cot(x) Now we can add because they have the same denominator.  We get:

(csc(x)-1+csc(x)+1)/(cot(x)) simplifying the top

(2csc(x))/(cot(x)) To simplify I use the inverse function definitions csc(x)=1/sin(x) and cot(x) = cos(x)/sin(x) to get

2*1/sin(x)*1/(cos(x)/sin(x)) = 2*1/sin(x)*sin(x)/cos(x) = 2/cos(x) = 2sec(x)

Which is what we wanted to prove

cot(x)/(csc(x)+1) + (csc(x)+1)/cot(x) = 2sec(x)