Question

How do you prove this?

Answer 1

GIVEN
(tan § + sec § - 1) / tan § - sec § + 1) = (1 + sin §) / cos §

SIMLEST FORM THEN LHS
[(sin§ / cos §) + (1 / cos §) - 1]
------------------------------------------
[(sin§ / cos §) - (1 / cos §) + 1]

LCM IS cos § FOR BOTH NUMERATOR AND DEMOMINATOR
= [sin § + 1 - cos §] / cos §
--------------------------------------------
[sin § - 1 + cos §] / cos §

= [sin § + ( 1 - cos §)] [sin § + (1 + cos §)]
---------------------------------------------------
[sin § - (1 + cos §)] [sin § + (1 + cos §)]

= [ sin § + 1 - cos §] [sin § + 1 + cos §]
---------------------------------------------------
[sin § - 1 + cos §)] [sin § + 1 + cos §]

= [sin^2 § + sin § + sin § cos § + sin § +
1 + cos § - sin § cos § - cos § - cos ^2 §]
—----------------------------------------------
[sin^2 § + sin § + sin § cos § - sin § -
1 - cos § + sin § cos § + cos § + cos ^2 §]

= ( sin^2 § - cos ^2 § + 2 sin § + 1)
---------------------------------------
(1 - 1 + 2sin § cos §)

= (sin^2 § - cos ^2 § + sin^2 § + cos ^2 § + 2sin §)
-----------------------------------------------------
(2sin § cos §)

= 2 sin § (sin § + 1)
-------------------------
2(sin § cos §)

= sin § + 1
--------------
cos §