Question

Answer the following:

If sec θ = `sqrt(2)` and `(3pi)/2 < theta < 2pi` then evaluate `(1 + tantheta + "cosec"theta)/(1 + cottheta - "cosec"theta)`

Answer 1

Given, sec θ = `sqrt(2)`

We know that,

tan²θ = sec²θ – 1

= `(sqrt(2))^2 - 1`

= 2 – 1 = 1

∴ tan θ = ± 1

Since `(3pi)/2 < theta < 2pi`,

θ lies in the 4^th quadrant.

∴ tan θ < 0

∴ tan θ = – 1

cot θ = `1/(tan theta)` = – 1

cos θ = `1/sec theta = 1/sqrt(2)`

tan θ = `sintheta/costheta`

∴ sin θ = tan θ cos θ

= `(-1)(1/sqrt(2))`

= `-1/sqrt(2)`

∴ cosec θ = `1/sin theta = -sqrt(2)`

∴ `(1 + tan theta + "cosec" theta)/(1 + cot theta - "cosec" theta)`

= `(1 - 1 - sqrt(2))/(1 - 1 + sqrt(2))`

= `(-sqrt(2))/sqrt(2)`

= – 1