How do you prove #(tan u + cot u)(cos u + sin u) = csc u + sec u#?

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Answer 1 · 2015-09-01T21:08:57

First, expand the Left Hand Side#("LHS")#

#"LHS"=(tan u + cot u)(cos u + sin u)#

#=tanucosu+tanusinu+cotucosu+cotusinu#

Recall that : #tanu=sinu/cosu#
and #" "cotu=cosu/sinu#

#=>"LHS"=sinu/cancel(cosu)*cancel(cosu)+sinu/cosu*sinu+cosu/sinu*cosu+cosu/cancel(sinu)*cancel(sinu)#

#=sinu+sin^2u/cosu+cos^2u/sinu+cosu#

#=sinu+cos^2u/sinu+cosu+sin^2u/cosu#

#=sin^2u/sinu+cos^2u/sinu+cos^2u/cosu+sin^2u/cosu#

#=(sin^2u+cos^2u)/sinu+(cos^2u+sin^2u)/cosu#

Recall again that : #sin^2u + cos^2u=1#

#=>"LHS"=1/sinu+1/cosu#

Recall that : #1/sinu=cscu#
and also #" "1/cosu=secu#

#=>"LHS"=color(blue)(cscu+secu)#

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