How do you prove #cot(theta)sec(theta)=csc(theta)#?

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Answer 1 · 2016-05-30T18:27:57

Applying the definition of #cot, sec, csc# and #tan#.

Recall the definition of these functions:

#cot(theta)=1/tan(theta)#
#sec(theta)=1/cos(theta)#
#csc(theta)=1/sin(theta)#

Then we want to prove

#cot(theta)sec(theta)=csc(theta)#

that is equivalent to

#1/tan(theta)1/cos(theta)=1/sin(theta)#

We recall that #tan(theta)=sin(theta)/cos(theta)#, consequently
#1/tan(theta)=cos(theta)/sin(theta)#.
I substitute in the previous equation

#1/tan(theta)1/cos(theta)=1/sin(theta)#

#cos(theta)/sin(theta)1/cos(theta)=1/sin(theta)#

#1/sin(theta)=1/sin(theta)#.