Establish the following identity: csc(theta) - cot(theta) = (sin(theta))/(1+cos(theta))?

Answer 1

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LHS: csc θ - cot θ =

1/sin θ - cos θ/sin θ =

(1 - cos θ)/sin θ

Multiply it by (1 + cos θ)/(1 + cos θ):

(1 - cos θ) (1 + cos θ) / (sin θ * (1 + cos θ)) =

(1 - cos^2 θ)/(sin θ * (1 + cos θ)) =

sin^2 θ/(sin θ * (1 + cos θ)) =

sin θ / (1 + cos θ) = sin θ / (1 + cos θ): RHS