Question

this ques is frm trigonometry
prove that cotθ-1+cosecθ ÷ cotθ+1-cosecθ = 1÷cosecθ-cotθ
               

Answer 1

let theta = Ф
(cotФ-1+cosecФ)/(cotФ+1-cosecФ)
(cosФ-sinФ+1)/(cosФ+sinФ-1)
(cosФ+1-sinФ)(cosФ+1+sinФ)/{(cosФ+sinФ-1)(cosФ+sinФ+1)}
{(cosФ+1)²-sin²Ф}/{(cosФ+sinФ)²-1²}
{cos²Ф+1+2cosФ-sin²Ф}/{cos²Ф+sin²Ф+2sinФcosФ+1}
{cos²Ф+cos²Ф+sin²Ф+2cosФ-sin²Ф}/{1+2sinФcosФ+1}
{2cos²Ф+2cosФ}/{2+2sinФcosФ}
(1+cosФ)/sinФ
(1+cosФ)(1-cosФ)/{(1-cosФ)sinФ}
(1-cos²Ф)/{(1-cosФ)sinФ}
sin²Ф/{(1-cosФ)sinФ}
sinФ/(1-cosФ)
1/(cosecФ-cotФ)
hence proved