How do you verify the identity #tanthetacsc^2theta-tantheta=cottheta#?

2 Answers
ProfLayton
Sep 10, 2016

Proof below

Explanation:

#tantheta * csc^2theta - tantheta#
#=sintheta/costheta * (1/sintheta)^2 - sintheta/costheta#
#=sintheta/costheta * 1/sin^2theta - sintheta/costheta#
#=1/(sinthetacostheta) - sintheta/costheta#
#=(1-sin^2theta)/(sinthetacostheta)#
#=cos^2theta/(sinthetacostheta)#
#=costheta/sintheta#
#=cottheta#

Note that #sin^2theta + cos^2theta = 1#, therefore #cos^2theta = 1- sin^2theta#

P dilip_k
Sep 10, 2016

#LHS=tantheta * csc^2theta - tantheta#

#=tantheta ( csc^2theta - 1)#

#=tantheta ( 1+cot^2theta - 1)#

#=tantheta * cot^2theta#

#=cottheta=RHS#

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