Question

How do you verify `(sec theta+csc theta)(cos theta-sin theta)=cot theta-tan theta`?

Answer 1

Recall the reciprocal identities:

`sectheta = 1/costheta`

`csctheta = 1/sintheta`

`cottheta = 1/tantheta`

Also, the quotient identities will be helpful

`tantheta = sintheta/costheta`

`cottheta = costheta/sintheta`

Now, simplify both sides:

`(1/costheta + 1/sintheta)(costheta - sin theta) = costheta/sintheta - sintheta/costheta`

`(sin theta + costheta)/(costhetasintheta)xx (costheta - sintheta) = (cos^2theta - sin^2theta)/(costhetasintheta`

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

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

Identity proved!!