Prove the following identity: (1-cos x)/(sin x)+ (sin x)/(1-cos x)=2csc x

hetjurnar02

hetjurnar02

Answered question

2022-09-02

Prove the following identity: 1 cos x sin x + sin x 1 cos x = 2 csc x

Answer & Explanation

Heidy Phelps

Heidy Phelps

Beginner2022-09-03Added 4 answers

L.H.S.
[ ( 1 cos x ) / sin x ] + [ sin x / ( 1 cos x ) ] = [ ( 1 cos x ) ( 1 cos x ) + ( sin x sin x ) ] / [ sin x ( 1 cos x ) ]....doingL.C.M.
= [ 1 + cos 2 x 2 cos x + sin 2 x ] / [ sin x ( 1 cos x ) ]
= [ 1 + cos 2 x + sin 2 x 2 cos x ] / [ sin x ( 1 cos x ) ]
= [ 1 + 1 2 cos x ] / [ sin x ( 1 cos x ) ] .......as sin 2 x + cos 2 x = 1
= [ 2 2 cos x ] / [ sin x ( 1 cos x ) ]
= 2 ( 1 cos x ) / [ sin x ( 1 cos x ) ]
= 2 / sin x
=2cosec x ......as 1 / sin x = c o s e c x
= R.H.S.
hence L.H.S. = R.H.S.= proved

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