Prove the following identities sec x/(cot x + tan x)=sin x

Question

Need help please

Answer 1

Before proving the identities, we need to know the basic identities.

 

sec(x) = 1 / cos(x)

cot(x) = cos(x) / sin(x)

tan(x) = sin(x) / cos(x)

 

Next is that we want to simplify the left side so that we can obtain the term on the right side.  If the left side equals to right side, then we have proven the identity.

 

 

(1 / cos(x)) / [(cos(x) / sin(x)) + (sin(x) / cos(x))] =

 

(1 / cos(x)) / [(cos²(x) + sin²(x)) / sin(x)cos(x)]

 

 

sin²(x) + cos²(x) = 1

 

 

(1 / cos(x)) / [1 / sin(x)cos(x)] =

 

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

 

sin(x)

 

This side equals the right side.