What is #cot theta + tantheta*sectheta # in terms of #sintheta #?

Question

1298 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-01-27T11:35:26

#((1-sin^2theta)^(3/2) +sin^2theta)/(sintheta(1-sin^2theta)#

You need to use the facts that #cot theta = 1/tan theta # and that #tan theta = sin theta/cos theta#

Then
#cot theta +tan theta * sec theta = cos theta/sin theta + sin theta / cos theta * 1/cos theta#

We know that #sin^2theta + cos^2theta = 1# so

#cos^2theta = 1 - sin^2theta# and

#costheta = sqrt(1-sin^2theta)#

Substituting these into the expression gives

#sqrt(1-sin^2theta)/sintheta +sintheta/(1-sin^2theta)#

#=((1-sin^2theta)^(3/2) +sin^2theta)/(sintheta(1-sin^2theta)#