Question

Question #5060b

Answer 1

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!