How do you verify the identity #(csc x - sin x)(sec x - cos x)(tan x + cot x) = 0#?

2 Answers
Gió
Apr 23, 2015

I am not sure that is equal to zero....I got #1#!
enter image source here
Have a look.

Alan P.
Apr 23, 2015

WARNING: This may not be the simplest way!
#(csc - sin)(sec - cos)(tan+cot)#
Let #s# represent #sin(x)#
#c# represent #cos(x)#
and #t# represent #tan(x)#

#(csc - sin)(sec - cos)(tan+cot)#
becomes
#(1/s - s)(1/c-c)(t+1/t)#

#=(1/(sc)-t - 1/t+sc)*(t+1/t)#

#((xx,|1/(sc),,-t,,-1/t,,+sc),( - , - , - , - , - , - , - , - - ),(1/t,|1/(sct),,-1,,-1/t^2,,+(sc)/t),(+t,|t/(sc),,-t^2,,-1,,sct) )#

Replacing #t# with #s/c#

we get
#1/(s^2) - 1 -c^2/s^2 +c^2 +1/c^2 -s^2/c^2 -1 +s^2#

#=(1-s^2)/(c^2) -2 +(c^2+s^2) +(1-c^2)/s^2#

#= 1#

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