Question

If `cot theta = 3/4`, prove that `sqrt((sec theta - cosec theta)/(sec theta + cosec theta)) = 1/sqrt7`

Answer 1

`cot theta = "𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒"/"𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒"`

Let x be the hypotenuse by applying Pythagoras theorem.

𝐴𝐶² = 𝐴𝐵² + 𝐵𝐶²

𝑥² = 16 + 9

`x^2 = 25 => x = 5`

`sec theta = (AC)/(BC) = 5/3`

`cosec theta = (AC)/(AB) = 5/4`

On substituting in equation we get

`sqrt((sec theta - cosec theta)/(sec theta + cosec theta)) = sqrt((5/3 - 5/4)/(5/3 + 5/4))`

`= sqrt(((20 - 15)/12)/((20 + 15)/12)) = sqrt(5/35) = 1/sqrt7`