Question

Verify the identity algebraically. This problem is very intriguing and awesome at the same time. It's wonderfully amazing!

1.) TAN³α-1/TAN α-1= TAN²α + TAN α + 1

Answer 1

For ease of typing, let tan a = x

so your question becomes,
prove
(x^3 - 1)/(x-1) = x^2 + x + 1

The numerator of the LS is a difference of cubes
LS = ( (x-1)(x^2 + x +1) )/(x-1)
= x^2 + x + 1
= LS

all done!