prove cosecѲ+cotѲ/cosecѲ-cotѲ=1+2cot2Ѳ+2cosecѲcotѲ

Question

this qiestions about trigonometry identities ..

Answer 1

Given (CosecѲ+cotѲ)/((CosecѲ-cotѲ) = 1+2Cot^2Ѳ+2CosecѲcotѲ

Left hand side identity = (CosecѲ+cotѲ)/((CosecѲ-cotѲ)

= [(1/SinѲ)+(CosѲ/SinѲ)]/[(1/SinѲ)-(CosѲ/SinѲ)]

= [(1+CosѲ)/SinѲ]/[(1-CosѲ)/SinѲ]

= (1+CosѲ)/(1-CosѲ)

Multiple with (1+CosѲ) to numarator and denominator.

= (1+CosѲ)^2/(1-Cos^2Ѳ)

= (1+Cos^2Ѳ+2CosѲ)/Sin^2Ѳ

= 1/Sin^2Ѳ+Cos^2Ѳ/Sin^2Ѳ+2CosѲ/Sin^2Ѳ

= Cosec^2Ѳ+Cot^2Ѳ+2(CosѲ/SinѲ)(1/SinѲ)

= Cot^2Ѳ+1+Cot^2Ѳ+2CotѲCosecѲ

= 1+2Cot^2Ѳ+2CosecѲCotѲ

= right hand side identity.