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

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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)#