The value of √1+cosθ/1-cosθ is

Question

The value of \(\sqrt{\frac{1+cosθ}{1-cosθ}}\) is

A. cot θ − cosec θ 

B. cosec θ + cot θ 

C^. cosec²θ + cot²θ 

D. (cot θ + cosec θ)²

Answer 1

Note: Since all the options involve the trigonometric ratios cosec θ and cot θ, so we divide the whole term (numerator as well as denominator) by sin θ. 

To find: \(\sqrt{\frac{1+cosθ}{1-cosθ}}\) 

Consider \(\sqrt{\frac{1+cosθ}{1-cosθ}}\) 

Dividing numerator and denominator by sin θ, we get

Rationalizing the term by multiplying it by \(\sqrt{ cosecθ + cotθ}\)

\(\sqrt{\frac{(cosecθ+cotθ)^2}{cosec^2θ-cot^2θ}}\)  

Now, as 1 + cot²θ = cosec²θ 

⇒ cosec²θ – cot²θ = 1

\(\sqrt{\frac{1+cosθ}{1-cosθ}}\) = \(\sqrt{\frac{(cosecθ+cotθ)^2}{cosec^2θ-cot^2θ}}\) 

= \(\sqrt{(cosec θ + cot θ)^2}\)

= cosec θ + cot θ