Prove the identity: csc x - cot x = sinx/1+cosx

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity csc x - cot x = sin x/(1+cos x) has to be proved.

Start from the left hand side

csc x - cot x

csc x = 1/ sin x and cot x = cos x/ sin x

=> 1/sin x - cos x/sin x

=> (1 - cos x)/sin x

=> (1 - cos x)(1 + cos x)/(sin x)*(1 + cos )

=> (1 - (cos x)^2)/(sin x)*(1 + cos )

=> (sin x)^2/(sin x)*(1 + cos )

=> sin x / (1 + cos )

which is the right  hand side.

This proves csc x - cot x = sin x/(1+cos x)

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