cos/sec x sin = csc - sin

Question

trigonometry identity question

Answer 1

Prove cos Θ/sec Θ sin Θ = csc Θ - sin Θ.

cos Θ/sec Θ sin Θ

=cos Θ / (1/cos Θ)(sin Θ)  note: [sec Θ = 1/cos Θ]

=cos²Θ/ sin Θ                   note:[cos Θ ÷ 1/cos Θ = cos Θ * cos Θ = cos²Θ]

=1 - sin²Θ/sin Θ                note:[cos²Θ = 1-sin²Θ]

=1/sin Θ - sin²Θ/sinΘ

=csc Θ - sin Θ                   note:[1/sin Θ = csc Θ]