What is the is the solution for this:

Question

Establish :

tan theta + cot theta
--------------------- = csc² theta
tan theta

Answer 1

tan theta + cot theta =

tan theta + 1 / tan theta =

tan theta * tan theta / tan theta + 1 / tan theta =

tan ^ 2 theta / tan theta + 1 / tan theta =

( tan ^ 2 theta + 1 ) / tan theta =

( sin ^ 2 theta / cos ^ 2 theta + cos ^ 2 theta ) / tan theta =

[ ( sin ^ 2 theta + cos ^ 2 theta ) / cos ^ 2 theta ] / tan theta =

( 1 / cos ^ 2 theta ) / tan theta =

1 / ( cos ^ 2 theta * tan theta ) =

1 / ( cos ^ 2 theta * sin theta / cos theta ) =

( 1 / cos theta * sin thheta )

1 / cos theta * 1 / sin theta =

sec theta * csc theta

( tan theta + cot theta ) / tan theta = csc theta

( sec theta * csc theta ) / tan theta = csc theta Divide both sides by csc theta

sec theta / tan theta = 1

1 / cos theta / ( sin theta / cos theta ) = 1

1 * cos theta / ( sin theta * cos theta ) = 1

1 / sin theta = 1

csc theta = 1

The solution is :

theta = pi / 2 + OR - 2 n * pi

where n is integer number