prove-cosec-2-cot-cosec-1-

Question Number 47836 by Aknabob1 last updated on 15/Nov/18

p r o v e {c o s e c}^{2} θ - c o t θ c o s e c θ = 1

Answered by $@ty@m last updated on 15/Nov/18

L H S = {c o s e c}^{2} θ - c o t θ c o s e c θ

= {c o s e c}^{2} θ - \frac{\cos θ}{\sin θ} \times \frac{1}{\sin θ}

= \frac{1}{\sin^{2} θ} - \frac{\cos θ}{\sin^{2} θ}

= \frac{1 - \cos θ}{\sin^{2} θ}

= \frac{1 - \cos θ}{1 - \cos^{2} θ}

= \frac{1}{1 + \cos θ}

\neq 1

T h e q u e s t i o n i s w r o n g .

M a y b e t y p o e r r o r .

P l . c h e c k

Commented by Aknabob1 last updated on 15/Nov/18

t h a n k s i a p p r e c i a t e

Answered by peter frank last updated on 15/Nov/18

cosec θ (cosec θ - \cot θ)

cosec θ (\frac{1}{\sin θ} - \frac{\cos θ}{\sin θ})

\frac{cosec θ}{\sin θ} (1 - \cos θ)

\frac{1 - \cos θ}{\sin^{2} θ} = \frac{1 - \cos θ}{1 - \cos^{2} θ}

= \frac{1 - \cos θ}{1 + \cos θ}

hence {c o s e c}^{2} θ - c o t θ c o s e c θ \neq 1

Commented by Aknabob1 last updated on 15/Nov/18

t h a n k s i a p p r e c i a t e