SOLUTION: When tan(theta) + sec(theta) = x show that sin(theta) = (x^2-1)/(x^2+1)

Given  show that  

We begin by proving the converse.  That is first we prove:

Given  show that 

[IOW, we start with what we have to prove, prove what was given,
and then we reverse the proof.]

Draw a right triangle containing angle theta with opposite
side x²-1 and hypotenuse x²+1, 



and we find what the adjacent side 
must equal using the Pythagorean theorem:




factor the left side as the difference of squares:






So we complete the right triangle with adjacent side 2x:



From that triangle we can prove what was given: 








So now that we have proved the converse of the problem,
we can reverse the proof:

-----------------------------------

Given ,
reversing the above steps:






From that and 
we can draw the right triangle:



Then from that right triangle, it follows that



Edwin