Question

Prove that cot theta - tan theta = 2cos^2theta - 1/sin theta cos theta

Answer 1

The key here is to convert both the tan(theta) and cot(theta) to sines and cosines.

The problem becomes: (I am skipping the theta's for now - makes typing faster. will put the back in before it is all over...)

sin cos
--- - ---
cos sin
------------- + 2 cos2(theta) = 1
sin cos
--- + ---
cos sin

Turn this complex fraction into a single fraction - get a common denominator for the top, and a common denominator for the bottom.

sin2 - cos2
---------------
sin cos
----------------- + 2 cos2(theta) = 1
sin2 + cos2
--------------
sin cos

Now, multiply the fraction on top and bottom by (sin cos) and you will no longer have a fraction of fractions, but a single fraction that looks like

sin2 - cos2
-------------- + 2 cos2(theta) = 1
sin2 + cos2

Remember the trig identity: sin2(theta) + cos2(theta) = 1, so the denominator of the fraction = 1.

Now what you have is

sin2(theta) - cos2(theta) + 2cos2(theta) = 1

The two "cos2" terms are like terms and can be combined, so the left side is now

sin2(theta) + cos2(theta) = 1

and the same identity as above makes the left side 1. So

1 = 1 and it is verified.