Question

If cosec θ = `("p" + "q")/("p" - "q")` (p ≠ q ≠ 0), then `|cot(π/4 + θ/2)|` is equal to ______.

Options

• `sqrt("p"/"q")`

• `sqrt("q"/"p")`

• `sqrt("pq")`

• pq

Answer 1

If cosecθ = `("p" + "q")/("p" - "q")` (p ≠ q ≠ 0), then `|cot(π/4 + θ/2)|` is equal to `underlinebb(sqrt("q"/"p"))`.

Explanation:

cosecθ = `("p" + 1)/("p" - q)`, sinθ = `("p" - "q")/("p" + "q")`

cosθ = ±`sqrt(1-sin^2theta)=sqrt(1-(("p"-"q")/("p"+"q"))^2)=(2sqrt"pq")/(("p"+"q"))`

`|cot(pi/4+theta/2)|=|("cot"pi/4"cot"theta/2-1)/("cot"pi/4+"cot"theta/2)|=|("cot"theta/2-1)/("cot"theta/2+1)|`

= `|("cot"theta/2-"sin"theta/2)/("cos"theta/2+"sin"theta/2)|`

on rationalizing denominator, we get

`|(("cos"theta/2-"sin"theta/2)/("cos"theta/2+"sin"theta/2))(("cos"theta/2"sin"theta/2)/("cos"theta/2+"sin"theta/2))|`

= `|costheta/("sin"^2theta/2+"cos"^2theta/2+2"sin"theta/2"cos"theta/2)|`

= `|costheta/(1+sintheta)|=|(2sqrt"pq"//("p"+"q"))/(1+(("p"-"q"))/("p"+  "q"))|`

= `sqrt"pq"/"p"`

= `sqrt("q"/"p")`