Question

How do I simplify the expression (tanˆ2Ø -1)cot Ø / sin Ø - cosØ to a sum of two trigonometric functions?

I know this has to be done using Trigonometric Identities, but I keep getting stuck. Thanks in advance for the help!

Answer 1

`sectheta+csctheta`

Explanation:

START:
`((tan^2Ø -1)cot Ø) / (sin Ø - cosØ)=`

`((1/cot^2theta -1)cot Ø) / (sin Ø - cosØ)=`

`(cottheta/cot^2theta -cottheta) / (sin Ø - cosØ)=`

`(1/cottheta -cottheta) / (sin Ø - cosØ)=`

`(tantheta -cottheta) / ((sinthetacostheta)/costheta - cos^2theta/costheta)=`

`(sin^2theta/(costhetasintheta) -cos^2theta/(sinthetacostheta)) / ((sinthetacostheta)/costheta - cos^2theta/costheta)=`

`((sin^2theta-cos^2theta)/(sinthetacostheta)) / ((sinthetacostheta- cos^2theta)/costheta)=`

`(sin^2theta-cos^2theta)/(sinthetacostheta) *costheta/(sinthetacostheta- cos^2theta)=`

`((sintheta+costheta)(sintheta-costheta))/(sinthetacostheta) *costheta/(costheta(sintheta-costheta)=`

`((sintheta+costheta)cancel(sintheta-costheta))/(sinthetacancelcostheta) *cancelcostheta/(costhetacancel(sintheta-costheta)=`

`(sintheta+costheta)/(sinthetacostheta)=`

`cscthetasectheta(sintheta+costheta)=`

`sectheta+csctheta`