Question

How do you verify `(cosX+sinX)/(cscX+secX) = (cosX)(sinX)`?

Answer 1

Using the definitions of secant and cosecant

• `csc(x) = 1/sin(x)`
• `sec(x) = 1/cos(x)`

We have

`(cos(x)+sin(x))/(csc(x)+sec(x)) = (cos(x)+sin(x))/(1/sin(x)+1/cos(x))`

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

`=cos(x)sin(x)*(cos(x)+sin(x))/((cos(x)sin(x))/sin(x)+(cos(x)sin(x))/cos(x))`

`=cos(x)sin(x)*(cos(x)+sin(x))/(cos(x)+sin(x))`

`=cos(x)sin(x)`

(Note that this identity is only true where secant and cosecant are defined, that is, where `sin(x)!=0` and `cos(x)!=0`)