proving 問題

L.H.S.
= sin2a tan a + cos2a cot a + 2 sin a cos a
= ............
= cosec a sec a


R.H.S.
= tan a + cot a
= (sin a/cos a) + (cos a/sin a)
= (sin2a + cos2a)/(sin a cos a)
= 1/(sin a cos a)
= cosec a sec a

Hence, L.H.S. = R.H.S.
=

2008-10-08 12:42:38 補充：
An alternative method:

L.H.S.
= .....
= 1 / (sin a cos a)
= (sin^2 a + cos^2 a) / (sin a cos a)
= (sin a/cos a) + (cos a/sin a)
= tan a + cot a
= R.H.S.

sin^2 a tan a + cos^2 a cot a + 2 sin a cos a = tan a + cot a

LHS = sin^2 a tan a + cos^2 a cot a + 2 sin a cos a
= sin^2 a tan a + cos^2 a cot a + sin a cos a + sin a cos a
= sina ( sina tana + cos a ) + cos a ( sina + cot a cos a)
= sina ( sina tana + sina / tana ) + cosa( cosa tana + cota cos a )
= sin^2a ( tan^2 a + 1) / tana + cos^a (tan^2 a + 1 ) /tana
= (tan^a + 1 ) /tana ( sin^2 a + cos^2 a )
= (tan^a + 1 )/tana
= tana + cota = RHS

2008-10-08 19:25:17 補充：
我還有一個威力驚人的方法:
LHS = sin^2 a tan a + cos^2 a cot a + 2 sin a cos a
= sin^2 a tan a + cos^2 a cot a + 2 sin a cos a - tan a - cot a + tan a + cot a
= tana ( sin^2 a - 1) + cota (cos^2 a - 1) + 2sinacosa + tan a + cot a

2008-10-08 19:25:27 補充：
= - tana cos^2 a - cota sin^2 a + 2sina cosa + tan a + cot a
= - sina cosa - sinacosa + 2sina cosa + tana + cota
= tana + cota