Question

Show that 1/cosec0-1 -1/cosec0+1 =2tan^2

Answer 1

$\frac{1}{ \csc(θ) - 1} - \frac{1}{ \csc(θ) + 1} \\ = \frac{ \csc(θ) + 1 - ( \csc(θ ) - 1) }{ { \csc }^{2}θ - 1} \\ = \\ as \: 1 + { \cot }^{2}θ = { \ \csc }^{2}θ\ \\ = \frac{2}{ { \cot }^{2}θ} \\ \\ as \: \cot(θ) = \frac{1}{ \tan(θ) } \\ therefore \\ \\ \frac{2}{ { \cot}^{2}θ} \\ = 2 \times { \tan}^{2} θ \\ = 2 { \tan}^{2} θ$

= RHS