Question

prove sin(90-theta)/cosec(90-theta)-cot theta = 1/sin theta​

Answer 1

sin(90-theta)/cosec(90-theta)-cot theta = 1/sin theta​

The correct question is,

sin(90-theta)/cosec(90-theta)-(90-theta) = 1/sin theta​

we have,

sin(90-theta) = cos theta

cosec(90-theta) = sec theta

LHS:

$= \dfrac{sin (90-\theta) }{csc (90 -\theta) cot (90 -\theta)}\\\\\\= \dfrac{cos \theta}{sec \theta - tan \theta}\\\\\\= \dfrac{cos \theta}{\frac{1}{cos \theta} - \frac{sin \theta}{cos \theta}}\\\\\\=\dfrac{cos^2 \theta}{1-sin \theta}\\\\\\=\dfrac{1-sin^2 \theta}{1-sin \theta}\\\\\\=\dfrac{(1-sin\theta)(1+sin \theta)}{1-sin\theta}\\\\= 1 + sin \theta$

RHS

Hence LHS = RHS

Hence the proof.