Question

How do you prove `(1+tany)/(1+coty)=secy/cscy`?

Answer 1

See below.

Explanation:

Apply the following identities:

•`tantheta = sintheta/costheta`

•`cottheta = 1/tantheta = 1/(sintheta/costheta) = costheta/sintheta`

•`sectheta = 1/costheta`

•`csctheta= 1/sintheta`

Start the simplification process on both sides.

`(1 + siny/cosy)/(1 + cosy/siny) = (1/cosy)/(1/siny)`

Put on a common denominator:

`((cosy + siny)/cosy)/((siny + cosy)/siny) = 1/cosy xx siny/1`

`(cosy + siny)/cosy xx siny/(siny + cosy) = siny/cosy`

`(cancel(cosy + siny))/cosy xx siny/(cancel(siny + cosy)) = siny/cosy`

`siny/cosy = siny/cosy`

Identity proved!!

Hopefully this helps!