Verify or disprove this identity --- cotx+tanx = (csc x)(sec x)?

Answer 1

recall..
cotx = cosx/ sinx
tanx = sinx / cosx

so we get .
cotx + tanx = cosx/sinx + sinx / cosx
make denominator same of both fraction
= cos^2x/ sinx cosx + sin^2x / sinx cosx
=(cos^2x + sin^2x ) / sinx cosx
=1/ sinx cosx (sin^2x + cos^2x = 1)
=1/sinx * 1/ cosx
=cscx secx

Answer 2

First, rewrite the left component: (tanx+cotx)(tanx+cotx)=csc^2(x)+sec^2(... Now, FOIL the left component out: tan^2(x)+2cotx*tanx+cot^2(x)=csc^2(x)+... cotx*tanx=a million because of the fact they are inverses of one yet another. this implies we are in a position to simplify to: tan^2(x)+cot^2(x)+2=csc^2(x)+sec^2(x) i'll break the two up right into a million and a million. Then i'll look on the left component as 2 products: [tan^2(x)+a million]+[cot^2(x)+a million]=csc^2(x)+sec... Now, go forward and discover uncomplicated denominators for all the products. For the 1st section the easy denominator is cos^2(x) and the appropriate piece's uncomplicated denominator is sin^2(x). this provides us: [(sin^2(x)+cos^2(x))/cos^2(x)]+[(cos^2... Now, sin^2(x)+cos^2(x)=a million, so we are in a position to simplify into: (a million/(cos^2(x)))+(a million/(sin^2(x)))=csc^2(x)... a million/cos^2(x)=sec^2(x) and a million/sin^2(x)=csc^2(x) So, we are in a position to simplify it into: sec^2(x)+csc^2(x)=csc^2(x)+sec^2(x) via the commutative regulation of addition, i will substitute the order of the left component to get: csc^2(x)+sec^2(x)=csc^2(x)+sec^2(x) Now, we've converted the left component into the appropriate component. this means that the identification has been shown.

Answer 3

cot x + tan x
= cos x / sin x + sin x / cos x
= (cos² x + sin²x) / (sin x.cos.x)
= 1 / ((sin x).(cos x))
= cosec x.sec x