Question #5060b

1 Answer
Mar 1, 2017

Use the trigonometric identity that states #sin^2theta+cos^2theta=1#

Explanation:

Typically, a mathematical proof follows a series of logical arguments to show that some theorem is true based on a set of axioms (one or more basic concepts that are assumed to be true).

If one can reach a logical conclusion based on an axiom, without committing any mathematical errors, then the theorem is "proven".

Let's start by assuming the following is true:

#sin^2theta+cos^2theta=1#

Subtracting #sin^2theta# from both sides, we get:

#cos^2theta=1-sin^2theta#

Let's replace #costheta# with #u# and #sintheta# with #v# for a moment:

#u^2=1-v^2#

Looking at the right hand side of the equation, it is a difference of squares, and so it can be factored into the following:

#u^2=(1-v)(1+v)#

Now, let's divide both sides by #(1-v)#

#u^2/(1-v)=(1+v)#

And, now, let's divide both sides by #u#

#u/(1-v)=(1+v)/u#

Let's separate the terms on the right hand side of the equation:

#u/(1-v)=1/u+v/u#

Finally, let's replace #u# and #v# with #costheta# and #sintheta#:

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

Simplifying, we get:

#costheta/(1-sintheta)=sectheta+tantheta#

Since we were able to logically show that the end result was true based on our initial assumption which was known to be true, it must be the case that #costheta/(1-sintheta)=sectheta+tantheta#

Note: This problem can also be worked in reverse!