How do you prove that #sectheta/tantheta=sintheta# is not an identity by showing a counterexample?

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Answer 1 · 2016-12-18T05:15:53

#tantheta/sectheta=sintheta#

Let's first simplify the left side and see what it actually equals, then we can come up with a counterexample:

#sectheta/tantheta=sintheta#

#(1/costheta)(1/tantheta)=sintheta#

#(1/costheta)(costheta/sintheta)=sintheta#

#(1/cancelcostheta)(cancelcostheta/sintheta)=sintheta#

#1/sintheta!=sintheta#

So we know that in the original equation, the left side of the equation equals the inverse of the right side. So we can write as a counterexample:

#tantheta/sectheta=sintheta#