Question

Show that `1+cot^2alpha/(1+cosecalpha)=cosec alpha`

Answer 1

LHS = `1+ (cot^(2)alpha)/(1+"cosec"alpha)=1+(cos^(2)alpha//sin^(2)alpha)/(1+1//sinalpha)` `[therefore cottheta=(costheta)/(sintheta)` and `"cosec"theta=1/(sintheta)]`
`=1+ (cos^(2)alpha)/(sinalpha(1+sin alpha))`
`=(sinalpha + 1)/(sinalpha(sinalpha+1))=1/sinalpha` `[therefore "cosec"theta=1/sintheta]`
`="cosec"alpha=RHS`