Question

If 12cosecθ = 13, find the value of `(sin^2θ  - cos^2θ) /(2sinθ  cosθ) xx (1)/tan^2θ`.

Answer 1

12cosecθ = 13

⇒ cosecθ = `(13)/(12)`

⇒ sinθ = `(12)/(13) = "Perpendicular"/"Hypotenuse"`

⇒ Base 
= `sqrt(("Hypotenuse")^2 - ("Perpendicular")^2`
= `sqrt((13)^2 - (12)^2`
= `sqrt(169 - 144)`
= `sqrt(25)`
= 5

cosθ = `"Base"/"Hypotenuse" = (5)/(13)`

tanθ= `"Perpendicular"/"Base" = (12)/(5)`

Now, `(sin^2θ  - cos^2θ) /(2sinθ  cosθ) xx (1)/tan^2θ`

= `((12/13)^2 - (5/13)^2)/(2(12/13)(5/13)) xx (1)/(12/5)^2`

= `(144/169 - 25/169)/(120/169) xx (25)/(144)`

= `(119)/(120) xx (25)/(144)`

= `(595)/(3456)`.