Question

If 5 cot θ = 12, find the value of : Cosec θ+ sec θ 

Answer 1

Consider the diagram below : 

5cot θ = 12
cot θ = `(12)/(5)`

i.e.`"base"/"perpendicular" = (12)/(5)`

Therefore if length of base = 12x, length of perpendicular = 5x

Since
base² + perpendicular² = hypotenuse²  ...[ Using Pythagoras Theorem]

(12x)² + (5x)² = hypotenuse²

hypotenuse² = 144x² + 25x² = 169x²

∴ hypotenuse = 13x

Now

cosec θ = `"hypotenuse"/"perpendicular" = (13x)/(5x) = (13)/(5)`

sec θ = `"hypotenuse"/"base" = (13x)/(12x) = (13)/(12)`

Therefore
cosec θ+sec θ

= `(13)/(5)+(13)/(12)`

= `(221)/(60)`

= `3(41)/(60)`