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

1 Answer
Bdub
Mar 30, 2017

see below

Explanation:

Use the identity #color(blue)(sin^2theta+cos^2theta=1# and solve for #sin^2 theta, and sin theta#. That is,

#color(blue)(sin^2 theta = 1-cos^2 theta#
#color(blue)(sin theta = +- sqrt(1-cos^2 theta)#

#sin^2 theta-sin theta+csc^2 theta=color(red)(sin^2theta-sin theta+1/sin^2 theta#

#=color(red)((1-cos^2 theta)-(+-sqrt(1-cos^2 theta))+1/(1-cos^2 theta)#

#=color(red)(((1-cos^2 theta)(1-cos^2 theta) -(1-cos^2 theta)(+- sqrt(1-cos^2 theta))+1)/(1-cos^2 theta)#

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