Question

How do you solve `\csc \theta -\sin \theta =\cos \theta \cot \theta`?

Answer 1

Note that this is an identity, not an equation, so we attempt to prove and not to solve.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Rewrite using the following fundamental identities:

•`cscbeta = 1/sinbeta`
•`cotbeta = 1/tanbeta = 1/(sinbeta/cosbeta) = cosbeta/sinbeta`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`1/sintheta - sin theta = costheta * costheta/sintheta`

`(1 - sin^2theta)/sintheta = cos^2theta/sintheta`

Simplify using `sin^2beta + cos^2beta = 1 -> cos^2beta = 1- sin^2beta`

`cos^2theta/sintheta = cos^2theta/sintheta`

Identity proved!

Hopefully this helps!