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

Question

13800 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-07T01:39:46

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!!

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