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# (1-cosectheta)(1-sectheta)(1+cosectheta)(1+costheta)=costheta

Sreevignesh
22 Points
4 years ago
I can't insert theta... So I am using x instead of theta
(1-cosec x ) ( 1-sec x) (1+cosec x ) ( 1+cos x)
=    (1-cosec x ) ( 1+ cosec x) (1- 1/cos x) ( 1+ cos x)
=     (1 - cosec ² x) [ ( cos x -1 )/ cos x] ( 1+ cos x ).  [ using (a+b)( a- b)= a²+b²
and taking LCM]
=    (- cot ² x) ×  [( cos ²x -1 ) /cos x].        [ multiply using ( a+b) ( a- b)=a²+b²
And   1+ cot ²x = cosec ²x]
=     (- cos ²x )/(sin ²x) × (- sin ²x)/ cos x.     [ cot x = cos x /sin x]
=     cos x                                                                          [ sin ²x+ cos ²x =1 ]
On cancelling , sin²x will be removed
Two - signs will become + and
Only one cos x will be there
Hence it is proved