Prove the following: cosecθ+cot θ-1/cosecθ+cot θ+1 = 1-sin θ/ cos θ

Question

Prove the following:

\(\frac{cosec\theta+cot\theta-1}{cosec\theta+cot\theta+1}=\frac{1-sin\theta}{cos\theta}\) 

cosecθ+cot θ-1/cosecθ+cot θ+1 = 1-sin θ/ cos θ

Answer 1

We know that, 

cot² θ = cosec² θ – 1 

∴ cot θ . cot θ = (cosec θ + 1)(cosec θ – 1)

∴ \(\frac {tan\theta}{sec\theta+1} =\frac{sec\theta-1}{tan\theta}\)

By the theorem on equal ratios, we get

 ∴ \(\frac {tan\theta}{sec\theta+1} =\frac{sec\theta-1}{tan\theta}\) = \(\frac{tan\theta+sec\theta-1}{sec\theta+1+tan\theta}\)

  ∴  = \(\frac{tan\theta+sec\theta-1}{sec\theta+1+tan\theta} = \frac {tan\theta}{sec\theta+1}\)

Alternate Method