Question

hii frnds.....
prove it 4 me.....
$cosec \: \alpha(1 + \cos \: \alpha )(cosec \: \alpha - cot \: \alpha ) = 1$

Answer 1

                                      ☺☺☺

Q. Prove that cosec a ( 1 + cos a ) ( cosec a - cot a ) = 1.

Solution: cosec a ( 1 + cos a ) ( cosec a - cot a ) = 1.     --------- equation 1


We know that cosec a = 1 / sin a  and cot a = cos a / sin a.By putting the value of  ( cosec a ) and  ( cot a  ) in equation 1.

 ( 1 / sin a ) ( 1 + cos a ) { ( 1 / sin a ) - ( cos a / sin a ) } = 1


                                            1 - cos a
( 1 / sin a ) ( 1 + cos a ) { ------------------- } = 1
                                              sin a

( 1 / sin² a ) ( 1 + cos a ) ( 1 - cos a ) = 1

( 1 / sin² a ) ( 1 - cos² a ) = 1.             ----------- equation 2

We know that ( 1 - cos² a ) = sin² a,

By putting the value of ( 1 - cos² a ) = sin² a in equation 2,

( 1 / sin² a ) x sin² a = 1

1 = 1.

 Hence, proved.

Note : Instead of alpha , ' a ' is being used.


                                       ☺☺☺

Answer 2

I am taking a instead of alpha

lhs=cosec a(1+ cos a) (cosec a-cot a)
(cosec a+cos a* cosec a) (cosec a -cot a)
(cosec a+cos a/sin a) (cosec a-cot a)
(cosec a+ cot a) (cosec a- cot a)
cosec2 a-cot2 a
1=rhs