How do you express #cos theta - csc^2 theta + sec theta # in terms of #sin theta #?

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Answer 1 · 2016-02-02T15:01:14

#(2sin^2theta -sin^4theta -sqrt(1-sin^2theta))/(sin^2thetasqrt(1-sin^2theta))#

Using trigonometric relationships we can say

#costheta - csc^2theta+sectheta#
#=costheta -1/sin^2theta +1/costheta#
#=(cos^2theta+1)/costheta - 1/sin^2theta#

We know that #cos^2theta =1-sin^2theta#

Substituting this into tthe expression gives
# (1-sin^2theta+1)/costheta -1/sin^2theta#

#=(sin^2theta(2-sin^2theta) - costheta)/(costhetasin^2theta)#

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

Substituting again gives

#(2sin^2theta -sin^4theta -sqrt(1-sin^2theta))/(sin^2thetasqrt(1-sin^2theta))#